NUMERALS    OR    COUNTERS? 

From  the  Margarita  Philosophica.     (See  page  67.) 


A,  Scrap -Book 


of 


Elementary  Mathematics 


Notes,  Recreations,  Essays 


By 

William  F.  White,  Ph.  D. 

State  Normal  School,  New  Paltz,  New  York 


Chicago 
The  Open  Court  Publishing  Company 

London  Agents 
Kegan  Paul,  Trench,  Trubner  &  Co.,  Ltd. 

1908 


Copyright  by 

The  Open  Court  Publishing  Co. 

1908. 


.  c    e     *  •  <•    •• 


CONTENTS. 

PAGE 

Preface 7 

The  two  systems  of  numeration  of  large  numbers 9 

Repeating   products 11 

Multiplication  at  sight:  a  new  trick  with  an  old  principle..  15 

A  repeating  table 17 

A  few  numerical  curiosities 19 

Nine '. 25 

Familiar  tricks  based  on  literal  arithmetic 27 

General  test  of  divisibility 30 

Test  of  divisibility  by  7 31 

Test  of  divisibility  by  7,  II,  and  13 32 

Miscellaneous  notes  on  number 34 

The  theory  of  numbers 34 

Fermat's   last   theorem 35 

Wilson's   theorem 35 

Formulas  for  prime  numbers 36 

A  Chinese  criterion  for  prime  numbers 36 

Are  there  more  than  one  set  of  prime  factors  of  a  num- 
ber?     2>7 

Asymptotic   laws 37 

Growth  of  the  concept  of  number 2>7 

Some  results  of  permutation  problems 27 

Tables 39 

Some  long  numbers 40 

How  may  a  particular  number  arise  ?   41 

Numbers  arising  from  measurement 43 

Decimals  as  indexes  of  degree  of. accuracy  of  measure..  44 

Some    applications 45 

Compound  interest 47 

If  the  Indians  hadn't  spent  the  $24 47 

3 


424KK4 


4         A<Stft\^£OOtK,OT  ELEMENTARY  MATHEMATICS. 

PAGE 

Decimal    separatrixes 49 

Present  trends  in  arithmetic 51 

Multiplication  and  division  of  decimals 59 

Arithmetic  in  the  Renaissance 66 

Napier's  rods  and  other  mechanical  aids  to  calculation.  .  .  69 

Axioms   in   elementary  algebra J2> 

Do  the  axioms  apply  to  equations  ? 76 

Checking  the  solution  of  an  equation 81 

Algebraic   fallacies 83 

Two  highest  common   factors 89 

Positive  and  negative  numbers 90 

Visual  representation  of  complex  numbers 92 

Illustration  of  the  law  of  signs  in  algebraic  multiplication.  97 

A  geometric  illustration 97 

From  a  definition  of  multiplication 98 

A  more  general  form  of  the  law  of  signs 99 

Multiplication  as  a  proportion 100 

Gradual  generalization  of  multiplication 100 

Exponents 101 

An  exponential  equation 102 

Two  negative  conclusions  reached  in  the  19th  century. .  . .  103 

The  three  parallel  postulates  illustrated 105 

Geometric    puzzles 109 

Paradromic   rings 117 

Division  of  plane  into  regular  polygons 118 

A  homemade  leveling  device 120 

"Rope   stretchers."    121 

The  three  famous  problems  of  antiquity 122 

The  circle  squarer's  paradox 126 

The  instruments  that  are  postulated 130 

The  triangle  and  its  circles 133 

•Linkages  and  straight-line  motion 136 

The  four-colors  theorem 140 

Parallelogram  of  forces 142 

A  question  of  fourth  dimension  by  analogy 143 


CONTENTS.  5 


PAGE 


Symmetry  illustrated  by  paper  folding 144 

Apparatus    to    illustrate    line    values    of    trigonometric    func- 
tions   146 

Sine 148 

Growth  of  the  philosophy  of  the  calculus 149 

Some   illustrations  of  limits. 152 

Law  of  commutation 154 

Equations  of  U.  S.  standards  of  length  and  mass 155 

The  mathematical  treatment  of  statistics 156 

Mathematical  symbols 162 

Beginnings  of  mathematics  on  the  Nile 164 

A  few  surprising  facts  in  the  history  of  mathematics 165 

Quotations  on  mathematics 166 

Autographs  of  mathematicians 168 

Bridges  and  isles,  figure  tracing,  unicursal  signatures,  lab- 
yrinths   I/O 

The  number  of  the  beast 180 

Magic   squares 183 

Domino  magic  squares 187 

Magic  hexagons 187 

The  square  of  Gotham 189 

A   mathematical   game-puzzle 191 

Puzzle   of  the   camels 193 

A   few  more  old-timers 194 

A  few  catch  questions 196 

Seven-counters  game 197 

To  determine  direction  by  a  watch 199 

Mathematical  advice  to  a  building  committee 201 

The  golden  age  of  mathematics 203 

The  movement  to  make  mathematics  teaching  more  con- 
crete   205 

The  mathematical  recitation  as  an  exercise  in  public  speak- 
ing   210 

The  nature  of  mathematical  reasoning 212 

Alice  in  the  wonderland  of  mathematics 218 


6         A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

PAGE 

Bibliographic   notes 234 

Mathematical  recreations 234 

Publication  of  foregoing  sections  in  periodicals 235 

Bibliographic  Index 236 

General  index 241 


PREFACE. 

Mathematics  is  the  language  of  definiteness,  the  ne- 
cessary vocabulary  of  those  who  know.  Hence  the  in- 
timate connection  between  mathematics  and  science. 

The  tendency  to  select  the  problems  and  illustra- 
tions of  mathematics  mostly  from  the  scientific,  com- 
mercial and  industrial  activities  of  to-day,  is  one  with 
which  the  writer  is  in  accord.  It  may  seem  that  in  the 
following  pages  puzzles  have  too  largely  taken  the 
place  of  problems.  But  this  is  not  a  text-book.  More- 
over, amusement  is  one  of  the  fields  of  applied  mathe- 
matics. 

The  author  desires  to  express  obligation  to  Prof. 
James  M.  Taylor,  LL.  D.,  of  Colgate  University 
(whose  pupil  the  author  was  for  four  years  and  after- 
ward his  assistant  for  two  years)  for  early  inspiration 
and  guidance  in  mathematical  study;  to  many  mathe- 
maticians who  have  favored  the  author  with  words  of 
encouragement  or  suggestion  while  some  of  the  sec- 
tions of  the  book  have  been  appearing  in  periodical 
form ;  and  to  the  authors  and  publishers  of  books  that 
have  been  used  in  preparation.  Footnotes  give,  in 
most  cases,  only  sufficient  reference  to  identify  the 
book  cited.  For  full  bibliographic  data  see  pages 
236-240.  Special  thanks  are  due  to  E.  B.  Escott, 
M.S.,  of  the  mathematics  department  of  the  University 
of  Michigan,  who  read  the  manuscript.  His  comments 
were  of  especial  value  in' the  theory  of  numbers.     Ex- 

7 


8         A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

tracts  from  his  notes  on  that  subject  (many  of  them 
hitherto  unpublished)  were  generously  placed  at  the 
disposal  of  the  present  writer.  Where  used,  mention 
of  the  name  will  generally  be  found  at  the  place. 
Grateful  acknowledgement  is  made  of  the  kindness 
and  the  critical  acumen  of  Mr.  Escott. 

The  arrangement  in  more  or  less  distinct  sections 
accounts  for  occasional  repetitions.  The  author  asks 
the  favor  of  notification  of  any  errors  that  may  be 
found. 

The  aim  has  been  to  present  some  of  the  most  inter- 
esting and  suggestive  phases  of  the  subject.  To  this 
aim,  all  others  have  yielded,  except  that  accuracy  has 
never  intentionally  been  sacrificed.  It  is  hoped  that 
this  little  book  may  be  found  to  possess  all  the  unity, 
completeness  and  originality  that  its  title  claims. 

The  Author. 

New  Paltz,  N.  Y.,  August,  1907. 


THE  TWO  SYSTEMS  OF  NUMERATION  OF 
LARGE  NUMBERS. 

What  does  a  billion  mean? 

In  Great  Britain  and  usually  in  the  northern  coun- 
tries of  Europe  the  numeration  of  numbers  is  by 
groups  of  six  figures  (106  =  million,  1012  =  billion,  1018 
=  trillion,  etc.)  while  in  south  European  countries  and 
in  America  it  is  by  groups  of  three  figures  ( 106  =  mil- 
lion, 10°  =  billion,  1012  =  trillion,  etc.).  Our  names 
are  derived  from  the  English  usage :  billion,  the  second 
power  of  a  million ;  trillion,  the  third  power  of  a  mil- 
lion ;  etc. 

As  the  difference  appears  only  in  such  large  num- 
bers, which  are  best  written  and  read  by  exponents, 
it  is  not  a  matter  of  practical  importance — indeed  the 
difference  in  usage  is  rarely  noticed — except  in  the 
case  of  billion.  This  word  is  often  heard ;  and  it 
means  a  thousand  million  when  spoken  by  one  half 
of  the  world,  and  a  million  million  in  the  mouths  of 
the  other  half. 

Billion.  "A  billion  does  not  strike  the  average 
mind  as  a  very  great  number  in  this  day  of  billion 
dollar  trusts,  yet  a  scientist  has  computed  that  at  10 :40 
a.  m.,  April  29, 1902,  only  a  billion  minutes  had  elapsed 
since  the  birth  of  Christ."  One  wonders  where  he 
obtained  the  data  for  such  accuracy,  but  the  general 
correctness  of  his  result  is  easily  verified.     "Billion" 


IO       A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

is  here  used  in  the  French  and  American  sense  (thou- 
sand million). 

An  English  professor  has  computed  that  if  Adam 
was  created  in  4004  B.  C.  (Ussher's  chronology),  and 
if  he  had  been  able  to  work  24  hours  a  day  contin- 
uously till  now  at  counting  at  the  rate  of  three  a 
second,  he  would  have  but  little  more  than  half  com- 
pleted the  task  of  counting  a  billion  in  the  English 
sense   (million  million). 


REPEATING  PRODUCTS. 

If  142857  be  multiplied  by  successive  numbers,  the 
figures   repeat   in    the   same   cyclic   order ;  1 

that  is,  they  read  around  the  circle  in  the  7  4 
margin  in  the  same  order,  but  beginning  at  5  2 
a  different  figure  each  time.  8 

2x142857=  285714 

3x  "  =  428571 

4x  "  =  571428 

5x  "  =  714285 

6x  "  =  857142 

7x  "  =  999999 

8x  "  =1142856. 

(When  we  attempt  to  put  this  seven -place  num- 
ber in  our  six-place  circle,  the  first  and  last  figures 
occupy  the  same  place.  Add  them,  and  we  still  have 
the  circular  order  142857.) 

9x142857=    1285713     (285714) 

10 x      "      =    1428570     (428571) 

11  x      "      =    1571427     (57142,$) 

23  x      "      =   3285711     (285714) 

89  x      "      = 12714273. 

(Again  placing  in  the  six  -  place  circular  order  and 

adding  figures  that  would  occupy  the  same  place,  or 

taking  the  12  and  adding  it  to  the  73,  we  have  714285.) 

356x142857  =  50857092    (adding  the   50  to 
the  092,   857142). 


12       A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

The  one  exception  given  above  (7x142857=999999) 
to  the  circular  order  furnishes  the  clew  to  the  identity 
of  this  "peculiar"  number:  142857  is  the  repetend  of 
the  fraction  1/1  expressed  decimally.  Similar  prop- 
erties belong  to  any  "perfect  repetend"  (repetend  the 
number  of  whose  figures  is  just  one  less  than  the  de- 
nominator of  the  common  fraction  to  which  the  circu- 
late is  equal).  Thus  »/„  =  .0588235294117647;  2x 
0588.  .  .  =  1176470588235294  (same  circular  order)  ; 
7  x  0588 ...  -  41 17647058823529 ;  while  17  x  0588 . .  . 
=9999999999999999.  So  also  with  the  repetend  of  V20, 
which  is  0344827586206896551724137931. 

It  is  easy  to  see  why,  in  reducing  \/p  (p  being  a 
prime)  to  a  decimal,  the  figures  must  begin  to  re- 
peat in  less  than  p  decimal  places ;  for  at  every  step 
in  the  process  of  division  the  remainder  must  be  less 
than  the  divisor.  There  are  therefore  only  p  -  1  dif- 
ferent numbers  that  can  be  remainder.  After  that 
the  process  repeats. 

7  W.lf  «  .14§  =*  .142?  =  .1428^  =  .14285| 

7        7  7  7  7  7 

=  .142857^  =  ... 

Hence  if  we  multiply  142857  by  3,  2,  6,  4,  5,  we  get 
the  repetend  beginning  after  the  1st,  2d,  3d,  4th,  5th 
figures  respectively. 

t>  —  1 
"If  a  repetend  contains  — —  digits,  all  the  multiples 

up  to  p  -  1  will  give  one  of  two  numbers  each  con- 

i>  —  \  1 

sisting  of y        digits.    Example:  —  ==  .076923 


REPEATING  PRODUCTS.  13 

1x76923=   76923        2x76923  =  153846 


3x 

U 

= 230769 

5x 

(t 

=  384615 

4x 

<* 

=  307692 

6x 

it 

=  461538 

9x 

a 

=  692307 

7x 

it 

=  538461 

10  x 

a 

=  769230 

8x 

a 

=  615384 

12  x 

a 

=  923076 

llx 

a 

=  846153" 
(Escott) 

"In  the  repetend  for  1/7,  if  we  divide  the  number 
into  halves,  their  sum  is  composed  of  9's,  viz.,  142  + 
857  =  999.  A  similar  property  is  true  of  the  repetend 
for  1/17  etc.  This  property  is  true  also  of  the  two 
numbers  obtained  from  1/13.  However,  when  we  find 
the  repetends  of  fractions  \/p  where  the  repetend  con- 

tains  only^--—  digits,  but  which  is  of  the  form  An  +  3, 

it  is  not  the  halves  of  the  numbers  which  are  comple- 
mentary, but  the  two  numbers  themselves.     Example: 

^  =  .032258064516129         ^-  =  .096774193548387 

SO 

—  =  .967741935483870 

Sum  =  .999999999999999"  (Escott) 

"A  useful  application  may  be  made  of  this  property 
of  repeating,  in  reducing  a  fraction  \/p  to  a  decimal. 
After  a  number  of  figures  have  been  found,  as  many 
more  may  be  found  by  multiplying  those  already  found 
by  the  remainder.  It  is,  of  course,  advantageous  to 
carry  on  the  work  until  a  comparatively  small  re- 
mainder has  been  found.  Example:  In  reducing  1/97 
to  a  decimal,  after  we  have  obtained  the  digits 
.01030927835  we  get  a  remainder  5.  Therefore,  from 
this  point  on  the  digits  are  the  same  as  those  of  1/97 
multiplied  by  5.     Multiplying  by  5  (or  dividing  by  2) 


14       A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

we  get  11  more  digits  at  once.  The  lengths  of  the 
periods  of  the  reciprocals  of  primes  have  been  deter- 
mined at  least  as  far  as  />  =  100,000."     (Escott.) 


MULTIPLICATION   AT    SIGHT:    A    NEW 
TRICK  WITH  AN  OLD  PRINCIPLE. 

This  property  of  repeating  the  figures,  possessed 
by  these  numbers,  enables  one  to  perform  certain  ope- 
rations that  seem  marvelous  till  the  observer  under- 
stands the  process.  For  example,  one  says :  "I  will 
write  the  multiplicand,  you  may  write  below  it  any 
multiplier  you  choose  of — say — two  or  three  figures, 
and  I  will  immediately  set  down  the  complete  product, 
writing  from  left  to  right."  He  writes  for  the  mul- 
tiplicand 142857.  Suppose  the  observers  then  write 
493  as  the  multiplier.  He  thinks  of  493  x  the  number 
as  493/7  =  703/7 ;  so  he  writes  70  as  the  first  figures 
of  the  product  (writing  from  left  to  right).  Now  3/7 
(i.  e.,  3  x  1/7)  is  thought  of  as  3  x  the  repetend,  and  it 
is  necessary  to  determine  first  where  to  begin  in  writ- 
ing the  figures  in  the  circular  order.  This  may  be 
determined  by  thinking  that,  since  3x7  (the  units 
figure  of  the  multiplicand)  =  21,  the  last  figure  is  1; 
therefore  the  first  figure  is  the  next  after  1  in  the 
circular  order,  namely  4.  (Or  one  may  obtain  the  4 
by  dividing  3  by  7.).  So  he  writes  in  the  product 
(after  the  70)  4285.  From  the  71  remaining,  the  70 
first  written  must  be  subtracted  (compare  the  expla- 
nation of  89x142857  given  above).  This  leaves  the 
last  two  figures  01,  and  the  product  stands  70428501. 
When  the  spectators  have  satisfied  themselves  by  ac- 
tual multiplication  that  this  is  the  correct  product,  let 

15 


l6       A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

us  suppose  that  they  test  the  "lightning  calculator" 
with  825  as  a  multiplier.  825/7=  117  6/7.  Write  117. 
6  x  7  =  42.  Next  figure  after  2  in  repetend  is  8.  Write 
857.  From  the  remaining  142  subtract  the  117.  Write 
025. 

Note  that  after  the  figures  obtained  by  division  (117 
in  the  last  example)  have  been  written,  there  remain 
just  six  figures  to  write,  and  that  the  number  first 
written  is  to  be  subtracted  from  the  six-place  number 
found  from  the  circular  order  (117  subtracted  from 
857142  in  the  last  example).  After  a  little  practice, 
products  may  be  written  in  this  way  without  hesita- 
tion. 

If  the  multiplier  is  a  multiple  of  7,  the  process  is 
even  simpler.  Take  378  for  multiplier.  378/7  =  54. 
Think  of  it  as  53  7/7.  Write  53.  7  x  the  repetend 
gives  six  nines.  Mentally  subtracting  53  from  999999, 
write,  after  the  53,  999946. 

This  trick  may  be  varied  in  many  ways,  so  as  not 
to  repeat.  (Few  such  performances  will  bear  repe- 
tition.) E.  g.,  the  operator  may  say,  "I  will  give  & 
multiplicand,  you  may  write  the  multiplier,  divide 
your  product  by  13,  and  I  will  write  the  quotient  as 
soon  as  you  have  written  the  multiplier."  He  then 
writes  as  multiplicand  1857141,  which  is  13x142857 
and  is  written  instantly  by  the  rule  above.  Now,  as 
the  13  cancels,  the  quotient  may  be  written  as  the 
product  was  written  in  the  foregoing  illustrations.  Of 
course  another  number  could  have  been  used  instead 
of  13. 


A  REPEATING  TABLE. 

Some  peculiarities  depending  on  the  decimal  nota- 
tion of  number.  The  first  is  the  sum  of  the  digits  in 
the  9's  table. 

9x  1=  9 

9x  2=  18;  1+8  =  9 

9x  3=  27;  2  +  7  =  9 

9x  4=  36;  3  +  6  =  9 


9x  9=  81: 
9  x  10  =  90 ; 
9x11=  99; 
9x12=108; 
9x13=117; 
etc. 


8+1=9 
9  +  0  =  9 
9  +  9=18;  1 
1+0+8=9 
1+1+7=9 


8  =  9 


The  following  are  given  by  Lucas*  in  a  note  entitled 

Multiplications  curieuscs : 

1x9  +  2=11 

12x9  +  3=111 

123x9  +  4=1111 

1234x9+5  =  11111 

12345x9  +  6=111111 

123456x9  +  7=1111111 

1234567x9  +  8  =  11111111 

12345678x9  +  9=111111111 

*  Recreations  Mathcmatiqucs,  IV,  232-3 ;  Thcone  des  Norn- 
brcs,  I,  8. 

■  7 


I  8       A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

9  x  9  +  7  =  88 

98  x  9  +  6  =  888 

987  x  9  +  5  =  8888 

9876  x  9  +  4  =  88888 

98765  x  9  +  3  =  888888 

987654  x  9  +  2  =  8888888 

9876543  x  9  + 1 =  88888888 

98765432  x  9  +  0  =  888888888 

1x8+1=9 

12x8  +  2  =  98 

123x8  +  3  =  987 

1234x8  +  4  =  9876 

12345x8  +  5=98765 

123456x8  +  6  =  987654 

1234567x8  +  7  =  9876543 

12345678x8  +  8  =  98765432 

123456789  x  8  +  9  =  987654321 

12345679x8  =  98765432 
12345679x9=111111111 

to  which  may,  of  course,  be  added 

12345679  x 18  =  222222222 
12345679x27  =  333333333 
12345679  x  36  =  \W\\\\\\ 
etc. 


A  FEW  NUMERICAL  CURIOSITIES.* 

112=121;  1112  =  12321;  11112  =  1234321;  etc. 

l  +  2  +  l  =  22;  l  +  2  +  3  +  2  +  l  =  32; 

l  +  2  +  3+4  +  3  +  2  +  l=42;  etc. 

-22X22.  333X333      ,  f 

m_l  +  2  +  l'     i^1-l  +  2  +  3  +  2  +  l'    etC'1 

The  following  three  consecutive  numbers  are  prob- 
ably the  lowest  that  are  divisible  by  cubes  other  than  1 : 

1375,  1376,  1377 
(divisible  by  the  cubes  of  5,  2  and  3  respectively). 

A  curious  property  of  37  and  41.  Certain  multiples 
of  37  are  still  multiples  of  37  when  their  figures 
are  permuted  cyclically :  259  =  7  x  37 ;  592  =  16  x  37  ; 
925  =  25  x  37.  The  same  is  true  of  185,  518  and  851 ; 
296,  629  and  962.  A  similar  property  is  true  of  mul- 
tiples of  41 :  17589  =  41  x  429;  75891  =  41  x  1851 ; 
58917  =  41x1437;  89175=41x2175;  91758  =  41  x 
2238. 

Numbers  differing  from  their  logarithms  only  in 
the  position  of  the  decimal  point.  The  determination 
of  such  numbers  has  been  discussed  by  Euler  and  by 
Professor  Tait.  Following  are  three  examples  of  a  list 
that  could  be  extended  indefinitely. 

*  Nearly  all  of  the  numerical  curiosities  in  this  section  were 
given  to  the  writer  by  Mr.  Escott. 

t  The  Monist,  1906 ;  XVI,  625. 

19 


20       A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

log  1.3712885742=  .13712885742 

log  237 . 5812087593  =  2 . 375812087593 

log  3550.2601815865  =  3.5502601815865 

Powers  having  same  digits 

Consecutive  numbers  whose  squares  have  the  same 
digits : 

132  =  169         1572  =  24649        9132  =  833569 
142  =  196        1582  =  24964        9142  =  835396 
Cubes  containing  the  same  digits: 

3453  =  41063625  3313  =  36264691 

3843  =  56623 104  4063  =  66923416 

4053  =  66430125 

A  pair  of  numbers  two  of  whose  powers  are  composed 

of  the  same  digits : 

322  =  1024  324  =  1048576 

492  =  2401  494  =  5764801 

Square  numbers  containing  tht  digits  not  repeated 
1.  Containing  the  nine  digits:* 

1 18262  =  139854276  203162  =  412739856 

123632  =  152843769  228872  =  523814769 

125432  =  157326849  230192  =  529874361 

146762  =  215384976  231782  =  537219684 

156812  =  245893761  234392  =  549386721 

159632  =  254817369  242372  =  587432169 

180722  =  326597184  242762  =  589324176 

190232  =  361874529  244412  =  597362481 

193772  =  375468129  248072  =  615387249 

195692  =  382945761  250592  =  627953481 

196292  =  385297641  255722  =  653927184 

*  Published  in  the  Mathematical  Magazine,  Washington,  D. 
C,  in  1883,  and  completed  in  L' Inter mediaire  des  Mathemati- 
ciens,  1897  (4:168). 


A    FEW    NUMERICAL   CURIOSITIES  21 

259412  =  672935481  27273'  -  743816529 

26409-  =  697435281  290342  =  8429731 56 

267 332  =  714653289  291062  =  847159236 

271292  =  735982641  303842  =  923187456 

2.  Containing  the  ten  digits  :f 

320432  =  1026753849  45624s  =  2081549376 
322862  =  1042385796  55446s  =  3074258916 
331442  =  1098524736  687632  =  4728350169 
351722  =  1237069584  839192  =  7042398561 
391472  =  1 532487609        990662  =  9814072356 

Arrangements  of  the  digits 

If  the  number  123456789  be  multiplied  by  all  the  in- 
tegers less  than  9  and  prime  to  9,  namely  2,  4,  5,  7,  8, 
each  product  contains  the  nine  digits  and  uses  each 
digit  but  once. 

Each  term  in  the  following  subtraction  contains 
each  of  the  nine  digits  once. 

987654321 
123456789 
864197532 

To  arrange  the  nine  digits  additively  so  as  to  make 
100: 


15 

56 

95i 

36 

8 

4|f 

47 

4 

100 

98 

3 

2 
100 

71 

29 

100 

other  solutions. 

See 

Fou: 

rrev 

and  Lucas. 

t L 'Inter mediaire  dcs  Mathematicians,  1907  (14:135) 


22       A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

To  arrange  the  ten  digits  additively  so  as  to  make 
100: 

50i  80H 

49ff  19* 

100  100 
Many  ways  of  doing  this  also. 

To  place  the  ten  digits  so  as  to  produce  each  of  the 
digits : 

'6?_97q_0  13i85_s 

31    485  02697 

62  485  34182 

31 X  970"  05697 

97062  -1  41832   ' 


48531  05976 

107469  25496 

35823"  03187 


=  8 


23184  _  57429  _       95742 

05796  ~  06381        ~  10638 

Lucas*  also  gives  examples  where  the  ten  digits  are 
used,  the  zero  not  occupying  the  first  place  in  a  number, 
for  all  of  the  ten  numbers  above  except  6,  which  is  im- 
possible. It  will  be  noticed  that,  in  the  example  given 
above  for  3,  the  digit  3  occurs  twice. 

The  nine  digits  arranged  to  form  a  perfect  cube: 

8__=  /   2_\3  8_         /   2  V      125    _/5\3 

32461759  ~  \  319/     24137569  ~  \  289/     438976     \76/ 

512        /8X3 


438976     \76, 
The  ten  digits  arranged  to  form  a  perfect  cube : 

9261        /91^3 
804357 

*  Thcoric  dcs  N ombres,  p.  40. 


\93/ 


A   FEW    NUMERICAL   CURIOSITIES.  23 

The  ten  digits  placed  to  give  an  approximate  value 
of  ?r: 

--fJlS-*MW»+ 

Fourier's  method  of  division*  by  a  number  of  two 
digits  of  which  the  units  digit  is  9.  Increase  the  di- 
visor by  1,  and  increase  the  dividend  used  at  each  step 
of  the  operation  by  the  quotient  figure  for  that  step. 
E.  g.,  43268^-29.    The  ordinary 

1492 
29)43268  29)43268 

29  1492 

142 
116 
266 
261 
58 
58 

arrangement  is  shown  at  the  left  for  comparison.  The 
form  at  the  right  is  all  that  need  be  written  in  Fou- 
rier's method.  To  perform  the  operation,  one  thinks 
of  the  divisor  as  30;  4—3,  (43-=-30,)  1  ;  write  the  1 
in  the  quotient  and  add  it  to  the  43  ;  44  -  30  =  14 ; 
14-f-3,  4 ;  etc.  The  reason  underlying  it  is  easily  seen. 
E.  g.,  at  the  second  step  we  have,  by  the  common 
method,  142-4x29.  By  Fourier's  method  we  have 
142  +  4  -  4  x  30.  The  addition  of  the  same  number 
(the  quotient  figure)  to  both  minuend  and  subtrahend 
does  not  affect  the  remainder. 

In  the  customary  method  for  the  foregoing  example 
one  practically  uses  30  as  divisor  in  determining  the 

*  Fourier,  p.  187. 


24       A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

quotient  figure  (thinking  at  the  second  step,  14^-3,  4). 
In  Fourier's  method  this  is  extended  to  the  whole 
operation  and  the  work  is  reduced  to  mere  short  divi- 
sion. 

So  also  in  dividing  by  19,  39,  49,  etc.  The  method 
is,  of  course,  not  limited  to  divisors  of  two  places,  nor 
to  those  ending  in  9.  It  may  be  used  in  dividing  by  a 
number  ending  in  8,  7  etc.  by  increasing  the  divisor  by 
2,  3  etc,  and  also  the  dividend  used  at  each  step  by  2,  3 
etc.  times  the  quotient  figure  for  that  step.  But  the 
advantage  of  the  method  lies  chiefly  in  the  case  first 
stated. 

"The  method  is  rediscovered  every  little  while  by 
some  one  and  hailed  as  a  great  discovery. " 


NINE. 

Curious  properties  of  the  number  nine,  and  numer- 
ical tricks  with  it,  are  given  and  explained  by  many 
writers ;  among  them  Dr.  Edward  Brooks,  in  his  Phi- 
losophy of  Arithmetic.  Of  all  such  properties,  perhaps 
the  most  practical  application  is  the  check  on  division 
and  multiplication  by  casting  out  nines,  the  Hindu  check 
as  it  is  called.  Next  might  come  the  bookkeeper's 
clue  to  inverted  numbers.  In  double-entry  book-keep- 
ing, if  there  has  been  inversion  (e.g.,  $4.83  written 
in  the  debit  side  of  one  account,  and  $4.38  in  the 
credit  side  of  another)  and  no  other  mistake,  the  trial 
balance  will  be  "off"  by  a  multiple  of  nine.  It  can 
also  be  seen  in  what  columns  the  transposition  was 
made. 

Recently  suggested,  and  of  no  practical  interest,  is 
another  property  of  the  "magic  number,"  easily  ex- 
plained, like  the  rest,  but  at  first  glance  curious:  in- 
vert the  figures  of  any  three-place  number ;  divide  the 
difference  between  the  original  number  and  the  in- 
verted number  by  nine ;  and  you  may  read 
the  quotient  forward. or  backward.     More-  845 

over  the  figure  that  occurs  in  the  quotient  548 

is  the  difference  between  the  first  and  last      9)297 
figures  of  the  number  taken.    Explanation :  33 

Let  a,  b,  c  be  the  hundreds,  tens,  units  fig- 
ures  respectively   of  any  three-place   number.     Then 
the  number  is  100a  +  10b  +  c,  and  the  number  inverted 
is   100c  +  10&  +  a. 

25 


26       A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

(100a  +  lQ{>+c)-(10Gc+10b  +  a)      99  (a-c) 
9  9 

The  product  of  1 1  and  any  one-place  number  will  have 
both  figures  alike,  and  may  be  read  either  way. 

Better  known  are  the  following  three — all  old  and 
all  depending  on  the  principle,  that  the  remainder, 
after  dividing  any  number  by  9,  is  the  same  as  the 
remainder  after  dividing  the  sum  of  its  digits  by  9. 

1.  Find  the  difference  betweeen  a  number  of  two 
figures  and  the  number  made  by  inverting  the  figures, 
conceal  the  numbers  from  me,  but  tell  me  one  figure 
of  the  difference.  I  will  tell  you  whether  there  is  an- 
other figure  in  the  difference,  and,  if  so,  what  it  is. 
(This  can  scarcely  be  repeated  without  every  spec- 
tator noticing  that  one  merely  subtracts  the  given 
figure  of  the  difference  from  9.) 

2.  Write  a  number  of  three  or  more  places,  divide 
by  9,  and  tell  me  the  remainder;  erase  one  figure,  not 
zero,  divide  the  resulting  number  by  9,  and  tell  me 
the  remainder.  I  will  tell  you  the  figure  erased 
(which  is,  of  course,  the  first  remainder  minus  the 
second,  or  if  the  first  is  not  greater  than  the  second, 
then  the  first  +9  -  the  second). 

3.  Write  a  number  with  a  missing  figure,  and  I 
will  immediately  fill  in  the  figure  necessary  to  make 
the  number  exactly  divisible  by  9.  (Suppose  728  57 
to  be  written.  Write  7  in  the  space ;  for  the  excess 
from  the  given  number  after  casting  out  9's  is  2,  and 
9-2  =  7.)  This  may  be  varied  by  undertaking  to  fill 
the  space  with  a  figure  that  shall  make  the  number 
divisible  by  nine  and  leaving  a  specified  remainder.* 

*  Adapted  from  Hooper,  I,  22 


FAMILIAR  TRICKS   BASED  ON  LITERAL 
ARITHMETIC. 

Besides  the  tricks  with  the  number  9,  there  are  many 
other  well-known  arithmetical  diversions,  most,  but  not 
all,  of  them,  depending  on  the  Arabic  notation  of  num- 
bers used.  Those  illustrated  in  this  section  are  spe- 
cially numerous,  can  be  "made  while  you  wait"  by  any 
one  with  a  little  ingenuity  and  an  elementary  knowl- 
edge of  algebra  (or,  more  properly,  of  literal  arith- 
metic) and,  when  set  forth,  are  transparent  the  mo- 
ment they  are  expressed  in  literal  notation.  They  are 
amusing  to  children,  and  it  is  no  wonder  that  the 
supply  of  them  is  perennial.  The  following  three  may 
be  given  as  fairly  good  types.  The  first  two  are  taken 
from  Dr.  Hooper's  book,  which  was  published  in  1774. 
Verbatim  quotation  of  them  is  made  in  order  to  pre- 
serve the  flavor  of  quaintness.  Only  the  explanation 
in  terms  of  literal  arithmetic  is  by  the  present  writer. 

1.  A  person  privately  fixing  on  any  number,  to  tell' 
him  that  number. 

After  the  person  has  fixed  on  a  number,  bid  him 
double  it  and  add  4  to  that  sum',  then  multiply  the 
whole  by  5  ;  to  the  product  let  him  add  12,  and  multi- 
ply the  amount  by  10.  From  the  sum  of  the  whole 
let  him  deduct  320,  and  tell  you  the  remainder,  from 
which,  if  you  cut  off  the  two  last  figures,  the  number 
that  remains  will  be  that  he  fixed  on. 


27 


28       A  SCRAP-BOOK  OF  ELEMENTARY   MATHEMATICS. 

Let  n  represent  any  number  selected.  The  first 
member  of  the  following  equality  readily  reduces  to 
n,  and  the  identity  explains  the  trick. 


{  [(2rc+4)5  +  12]l0-320;  -rl00  =  «. 

2.  Three  dice  being  thrown  on  a  table,  to  tell  the 
number  of  each  of  them,  and  the  order  in  which  they 
stand. 

Let  the  person  who  has  thrown  the  dice  double  the 
number  of  that  next  his  left  hand,  and  add  5  to  that 
sum;  then  multiply  the  amount  by  5,  and  to  the 
product  add  the  number  of  the  middle  die ;  then  let 
the  whole  be  multiplied  by  10,  and  to  that  product 
add  the  number  of  the  third  die.  From  the  total  let 
there  be  subtracted  250,  and  the  figures  of  the  number 
that  remains  will  answer  to  the  points  of  the  three 
dice  as  they  stand  on  the  table. 

Let  x,  y,  z  represent  the  numbers  of  points  shown 
on  the  three  dice  in  order.  Then  the  instructions, 
expressed  in  symbols,  give 

[(2„r  +  5)5  +  :y]10  +  xr-250. 

Removing  signs  of  grouping,  we  have 

lOO.r  +  lOy  +  s, 

the  number  represented  by 'the  three  digits  x,  y,  z  in 
order. 

3.  "Take  the  number  of  the  month  in  which  you 
were  born  (1  for  January,  2  for  February,  etc.), 
double  it ;  add  5  ;  multiply  by  50 ;  add  your  age  in 
years;  subtract  365;  add  115.  The  resulting  number 
indicates  your  age — month  and  years."  E.  g.,  a  per- 
son 19  years  old  and  born  in  August  (8th  month) 
would  have,  at  the  successive  stages  of  the  operation, 


FAMILIAR  TRICKS.  29 

8,  16,  21,  1050,  1069,  704;  and  for  the  final  number, 
819  (8  for  August,  19  for  the  years). 

If  we  let  m  represent  the  number  of  the  month, 
and  y  the  number  of  years,  we  can  express  the  rule 
as  a  formula: 

{2m  +  5)50  +  y-  365  +  115, 

which  simplifies  to 

100m  +  y, 

the  number  of  hundreds  being  the  number  of  the 
month,  and  the  number  expressed  by  the  last  two 
digits  being  the  number  of  years. 


GENERAL  TEST  OF  DIVISIBILITY.* 

Let  M  represent  any  integer  containing  no  prime 
factor  that  is  not  a  factor  of  10  (that  is,  no  primes 
but  5  and  2).  Then  1/M  expressed  decimally  is 
terminate.  Call  the  number  of  places  in  the  decimal 
m.  Let  N  represent  any  prime  except  5,  2,  1.  Then 
the  reciprocal  of  N  expressed  decimally  is  a  circulate. 
Call  the  number  of  places  in  the  repetend  n. 

1.  The  remainder  obtained  by  dividing  any  integer,. 
I,  by  M  is  the  same  as  that  obtained  by  dividing  the. 
number  represented  by  the  last  (right-hand)  m  digits 
of  I  by  M.     If  the  number  represented  by  those  m 
digits  is  divisible  by  M,  I  is  divisible  by  M,  and  not 
otherwise. 

2.  The  remainder  obtained  by  dividing  I  by  N  is 
the  same  as  that  obtained  by  dividing  the  sum  of  the 
numbers  expressed  by  the  successive  periods  of  n 
digits  of  I  by  N.  If  that  sum  is  divisible  by  N,  I  is 
divisible  by  N,  and  not  otherwise.  This  depends 
on  Fermat's  theorem,  that  P^  ~ '  -  1  is  divisible  by 
p  when  p  and  P  are  prime  to  each  other. 

3.  If  a  number  is  composite  and  contains  a  prime 
factor  other  than  5  and  2,  the  divisibility  of  I  by  it 
may  be  tested  bv  testing  with  the  factors  separately 
by    (1)    and    (2)'. 

*  Divisible  without  remainder  is  of  course  the  meaning  of 
"divisible"  in  such  a  connection. 


30 


GENERAL  TEST  OF  DIVISIBILITY.  3 1 

Thus  it  is  possible  to  test  the  divisibility  of  any  in- 
teger by  any  other  integer.  This  is  usually  of  only 
theoretic  interest,  as  actual  division  is  preferable.  But 
in  the  case  of  2,  3,  4,  5,  6,  8,  9  and  10  the  test  is  easy 
and  practical.  A  simple  statement  of  it  for  each  of 
these  particular  cases  is  found  in  almost  any  arith- 
metic. 

For  11a  test  slightly  easier  than  the  special  appli- 
cation of  the  general  test  is  usually  given.  That  is, 
subtract  the  sum  of  the  even-numbered  digits  from  the 
sum  of  the  odd-numbered  digits  (counting  from  the 
right)  and  add  11  to  the  minuend  if  smaller  than  the 
subtrahend.  The  result  gives  the  same  remainder 
when  divided  by  11  as  the  original  number  gives. 
The  original  number  is  divisible  by  11  if  that  result  is, 
and  not  otherwise.  These  remainders  may  be  used 
in  the  same  manner  as  the  remainders  used  in  casting 
out  the  nines,  but  are  not  so  conveniently  obtained. 

Test  of  divisibility  by  f.  No  known  form  of  the  gen- 
eral test  in  this  case  is  as  easy  as  actually  dividing  by  7. 
From  the  point  of  view  of  theory  it  may  be  worth 
noticing  that,  as  7's  reciprocal  gives  a  complementary 
repetend,  the  general  test  admits  of  variety  of  form.* 
Let  us  consider,  however,  the  direct  application. 

Since  the  repetend  has  6  places,  the  test  for  divisi- 
bility by  7  is  as  follows:  A  number  is  divisible  by  7 
if  the  sum  of  the  numbers  represented  by  the  suc- 
cessive periods  of  6  figures  each  is  divisible  by  7,  and 
not  otherwise ;  e.  g., 

Given  the  number  26,436,080,216,581 

*  A  chapter  of  Brooks's  Philosophy  of  Arithmetic  is  devoted 
to  divisibility  by  7. 


32       A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

216581 
436080 

26 

7)652687 
93241 

No  remainder;  therefore  the  given  number  is  divi- 
sible by  7. 

Test  of  divisibility  by  J,  n  and  JJ  at  the  same  time.* 
Since  7x  11  x  13  =  1001,  divide  the  given  number  by 
1001.  If  the  remainder  is  divisible  by  7,  11,  or  13, 
the  given  number  is  also,  and  not  otherwise. 

To  divide  by  1001,  subtract  each  digit  from  the 
third  digit  following.  An  inspection  of  a  division  by 
1001  will  show  why  this  simple  rule  holds.  The 
method  mav  be  made  clear  by  an  example,  4,728,350,169 
---1001. 

4728350169 
472^626543 
3 

Quotient,  4723626 ;  remainder,  543 

The  third   digit  before  the  4  being  0    (understood), 
x  write  the  difference,  4,  beneath  the  4.     Similarly  for 
7   and   2.     8-4  =  4    (which    for   illustration   is    here 
written  beneath  the  8).     We  should  next  have  3-7. 
'phis  changes  the  4  just  found  to  3,  and  puts  6  under 
the7    original   3    (that   is,   83-47  =  36).      5-2,   0-3 
(always  subtracting  from  a  digit  of  the  original  num- 
/     ber  the  third  digit  to  the  left  in  the  difference,  or  lower, 
number)  ,1-6,  etc.    Making  the  corrections  mentally 
we  have  the  number  as  written.    The  number  repre- 
sented by  the  last  three  digits,  543,  is  the  remainder 

*  This  was  given  to  the  author  by  Mr.  Escott,  who  writes : 
"I  have  never  seen  it  published,  but  it  is  so  simple  that  it  would 
be  surprising  if  it  had  not  been/' 


GENERAL  TEST  OF  DIVISIBILITY.  33 

after  dividing  the  given  number  by  1001,  and  the 
number  represented  by  the  other  digits,  4723626,  is 
the  quotient.  With  a  little  practice,  this  method  can 
be  applied  rapidly  and  without  making  erasures.  The 
remainder,  543,  which  alone  is  needed  for  the  test, 
may  also  be  obtained  by  subtracting  the  sum  of  the 
even-numbered  periods  of  three  figures  each  in  the 
original  number  from  the  sum  of  the  odd-numbered 
periods.  A  rapid  method  of  obtaining  the  remainder 
thus  is  easily  acquired  ;  but  the  way  illustrated  above 
is  more  convenient. 

However  obtained,  the  remainder  is  divisible  or  not 
by  7,  11  or  13,  according  as  the  given  number  is 
divisible  or  not.  (Here  543  is  not  divisible  by  7,  11 
or  13 ;  therefore  4728350169  is  not  divisible  by  either 
of  them.)  The  original  number  is  thus  replaced,  for 
the  purpose  of  investigation,  by  a  number  of  three 
places  at  most.  As  this  tests  for  three  common  primes 
at  once,  it  is  convenient  for  one  factoring  large  num- 
bers without  a  factor  table. 


MISCELLANEOUS  NOTES  ON   NUMBER. 

The  theory  of  numbers  has  been  called  a  "neglected 
but  singularly  fascinating  subject. "*  " Magic  charm" 
is  the  quality  ascribed  to  it  by  the  foremost  mathe- 
matician of  the  nineteenth  century .f  Gauss  said  also: 
"Mathematics  the  queen  of  the  sciences,  and  arithmetic 
[i.  e.,  theory  of  numbers]  the  crown  of  mathematics." 
And  he  was  master  of  the  sciences  of  his  time.  "While 
it  requires  some  facility  in  abstract  reasoning,  it  may 
be  taken  up  with  practically  no  technical  mathematics,  is 
easily  amenable  to  numerical  exemplifications,  and 
leads  readily  to  the  frontier.  It  is  perhaps  the  only 
branch  of  mathematics  where  there  is  any  possibility 
that  new  and  valuable  discoveries  might  be  made  with- 
out an  extensive  acquaintance  with  technical  mathe- 
matics. "% 

An  interesting  exercise  in  higher  arithmetic  is  to 
investigate  theorems  and  the  established  properties  of 
particular    numbers    to    determine    which    have    their 

*  Ball,  Hist.,  p.  416. 

t  "The  most  beautiful  theorems  of  higher  arithmetic  have 
this  peculiarity,  that  they  are  easily  discovered  by  induction, 
while  on  the  other  hand  their  demonstrations  lie  in  exceeding 
obscurity  and  can  be  ferreted  out  only  by  very  searching  in- 
vestigations. It  is  precisely  this  which  gives  to  higher  arith- 
metic that  magic  charm  which  has  made  it  the  favorite  science 
of  leading  mathematicians,  not  to  mention  its  inexhaustible 
richness,  wherein  it  so  far  excels  all  other  parts  of  pure 
mathematics."     (Gauss;  quoted  by  Young,  p.  155.) 

X  Young,  p.  155. 

34 


MISCELLANEOUS    NOTES   ON    NUMBER.  35 

origin  in  the  nature' of  number  itself  and  which  are 
due  to  the  decimal  scale  in  which  the  numbers  are 
expressed. 

Fer  mat's  last  theorem.  Of  the  many  theorems  in 
numbers  discovered  by  Fermat,  nearly  all  have  since 
been  proved.  A  well-known  exception  is  sometimes 
called  his  "last  theorem."  It  "is  to  the  effect  that  no 
integral  values  of  x,  y,  s  can  be  found  to  satisfy  the 
equation  xn  +  yn  -  zn,  if  n  is  an  integer  greater  than  2. 
This  proposition  has  acquired  extraordinary  celebrity 
from  the  fact  that  no  general  demonstration  of  it  has 
been  given,  but  there  is  no  reason  to  doubt  that  it  is 
true."*  It  has  been  proved  for  special  cases,  and  proved 
generally  if  certain  assumptions  be  granted.  Fermat 
asserted  that  he  had  a  valid  proof.  That  may  yet  be 
re-discovered ;  or,  more  likely,  a  new  proof  will  be 
found  by  some  new  method  of  attack.  "Interest  in 
problems  connected  with  the  theory  of  numbers  seems 
recently  to  have  flagged,  and  possibly  it  may  be  found 
hereafter  that  the  subject  is  approached  better  on 
other  lines. "f 

Wilson's  theorem  may  be  stated  as  follows:  If  p  is 
a  prime,  1  +\p  -  1  is  a  multiple  of  p.  This  well-known 
proposition  was  enunciated  by  Wilson,^  first  published 

*  Ball,  Recreations,  p.  2>7- 

f  Ball,  Hist.,  p.  469. 

X  As  he  was  not  a  professional  mathematician,  but  little 
mention  of  him  is  made  in  histories  of  the  subject.  The  fol- 
lowing items  may  be  of  interest.  They  are  from  De  Morgan's 
Budget  of  Paradoxes,  p.  132-3.  John  Wilson  (1741-1793)  was 
educated  at  Cambridge.  While  an  undergraduate  he  "was 
considered  stronger  in  algebra  than  any  one  in  the  University, 
except  Professor  Waring,  one  of  the  most  powerful  algebra- 
ists of  the  century."  Wilson  was  the  senior  wrangler  of  1761. 
He  entered  the  law,  became  a  judge,  and  attained  a  high  repu- 
tation. 


^6       A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

by  Waring  in  his  Meditationes  Algebraic®,  and  first 
proved  by  Lagrange  in  1771. 

Formulas  for  prime  numbers.  "It  is  easily  demon- 
strated that  no  rational  algebraic  formula  can  always 
give  primes.  Several  remarkable  expressions  have 
been  found,  however,  which  give  a  large  number  of 
primes  for  consecutive  values  of  x.  Legendre  gave 
2x2  +  29,  which  gives  primes  for  x  -  0  to  28,  or  for 
29  values  of  x.  Euler  found  x2  +  x  +  A\,  which  gives 
primes  for  x  -  0  to  39,  i.  e.,  40  values  of  x.  I  have 
found  6x2+6x+3l,  giving  primes  for  29  values  of  x  ;  and 
3x2  +  3x  +  23,  giving  primes  for  22  values  of  x.  These 
expressions  give  different  primes.  We  can  transform, 
them  so  that  they  will  give  primes  for  more  values  of 
x,  but  not  different  primes.  For  instance,  in  Euler's 
formula  if  we  replace  x  by  x  -  40,  we  get  x2  -  79 x 
+  1601,  which  gives  primes  for  80  consecutive  values 
of  xT     (Escott.) 

A  Chinese  criterion  for  prime  numbers.  With  ref- 
erence to  the  so-called  criterion,  that  a  number  p  is 
prime  when  the  condition,  that  2^  ~ l  -  1  be  divisible  by 
p,  is  satisfied,  Mr.  Escott  makes  the  following  inter- 
esting comment : 

"This  is  a  well-known  property  of  prime  numbers 
(Fermat's  Theorem)  but  it  is  not  sufficient.  My 
attention  was  drawn  to  the  problem  by  a  question  in 
L'Intermediaire  des  Mathematiciens,  which  led  to  a 
little  article  by  me  in  the  Messenger  of  Mathematics. 
As  the  smallest  number  which  satisfies  the  condition 
and  which  is  not  prime  is  341,  and  to  verify  it  by 
ordinary  arithmetic  (not  having  the  resources  of  the 
Theory  of  Numbers)  would  require  the  division  of 
2:H0-  1  by  341,  it  is  probable  that  the  Chinese  obtained 
the  test  bv  a  mere  induction." 


MISCELLANEOUS    NOTES    ON     NUMBER.  37 

Are  there  more  than  one  set  of  prime  factors  of  a 
number?  Most  text-books  answer  no ;  and  this  an- 
swer is  correct  if  only  arithmetic  numbers  are  con- 
sidered. But  when  the  conception  of  number  is  ex- 
tended to  include  complex  numbers,  the  proposition, 
that  a  number  can  be  factored  into  prime  factors  in 
only   one   wav,   ceases   to   hold.      E.    g.,   26  =  2x13  = 

(5+V-l)(5-V-l). 

Asymptotic  laws.  This  happily  chosen  name  de- 
scribes "laws  which  approximate  more  closely  to  ac- 
curacy as  the  numbers  concerned  become  larger."  * 
Legendre  is  among  the  best-known  names  here.  One 
of  the  most  celebrated  of  the  original  researches  of 
Dirichlet,  in  the  middle  of  the  last  century,  was  on 
this  branch  of  the  theory  of  numbers. 

Growth  of  the  concept  of  number,  from  the  arith- 
metic integers  of  the  Greeks,  through  the  rational 
fractions  of  Diophantus,  ratios  and  irrationals  recog- 
nized as  numbers  in  the  sixteenth  century,  negative 
versus  positive  numbers  fully  grasped  by  Girard  and 
Descartes,  imaginary  and  complex  by  Argand,  Wessel, 
Euler  and  Gauss, f  has  proceeded  in  recent  times  to 
new  theories  of  irrationals  and  the  establishing  of  the 
continuity  of  numbers  without  borrowing  it  from 
space.J 

Some  results  of  permutation  problems.  The  formu- 
las for  the  number  of  permutations,  and  the  number 
of  combinations,  of  n  dissimilar  things  taken  r  at  a 
time  are  given  in  every  higher  algebra.  The  most 
important  may  be  condensed  into  one  equality: 

nVr  =  n{n-l)  (n-2)  .  .  .(;,-r+l)  =  T-^-  =  "Cr  \r 

\n  —  r  — 

*  Ball,  Hist.,  p.  464.  t  See  p.  94. 

t  See   Cajori's  admirable   summary.  Hist,   of  Math.,  p.   372. 


38       A  SCRAP-COOK  OF  ELEMENTARY   MATHEMATICS. 

There  are  3,979,614,965,760  ways  of  arranging  a 
set  of  28  dominoes  (i.  e.,  a  set  from  double  zero  to 
double  six)   in  a  line,  with  like  numbers  in  contact.t 

"Suppose  the  letters  of  the  alphabet  to  be  wrote  so 
small  that  no  one  of  them  shall  take  up  more  space 
than  the  hundredth  part  of  a  square  inch :  to  find 
how  many  square  yards  it  would  require  to  write  all 
the  permutations  of  the  24  letters  in  that  size. "J  Dr. 
Hooper  computes  that  "it  would  require  a  surface 
18620  times  as  large  as  that  of  the  earth  to  write  all 
the  permutations  of  the  24  letters  in  the  size  above 
mentioned." 

Fear  has  been  expressed  that  if  the  epidemic  of 
organizing  societies  should  persist,  the  combinations 
and  permutations  of  initial  letters  might  become  ex- 
hausted. We  have  F.A.A.M.,  I.O.O.F.,  K.M.B.,  K.P., 
I.O.G.T.,  W.C.T.U.,  Y.M.C.A.,  Y.W.C.A.,  A.B.A., 
A.B.S.,  A.C.M.S.,  etc.,  etc.  An  almanac  names  more 
than  a  hundred  as  "prominent  in  New  York  City/* 
and  its  list  is  exclusive  of  fraternal  organizations,  of 
which  the  number  is  known  to  be  vast.  Already  there 
are  cases  of  two  societies  having  names  with  the 
same  initial  letters.  But  by  judicious  choice  this  can 
long  be  avoided.  Hooper's  calculation  supposed  the 
entire  alphabet  to  be  employed  in  every  combination. 
Societies  usually  employ  only  2,  3  or  4  letters.  And 
a  letter  may  repeat,  as  the  A  in  the  title  of  the  A.L.A. 
or  of  the  A.A.A.  The  present  problem  is  therefore 
different  from  that  above.  The  number  of  permuta- 
tions of  26  letters  taken  two  at  a  time,  the  two  being 
not  necessarily  dissimilar,  is  262 ;  three  at  a  time,  263 ; 

t  Ball,  Recreations,  p.  30,  citing  Reiss,  Annali  di  matcmatica. 
Milan,  Nov.  1871. 

t  Hooper,  I,  59. 


MISCELLANEOUS    NOTES    ON    NUMBER.  39 

etc.  As  there  is  occasionally  a  society  known  by  one 
letter  and  occasionally  one  known  by  five,  we  have 

261  +  262  +  26:!  +  264  +  26"  =  12,356,630. 

This  total  of  possible  permutations  is  easily  beyond 
immediate  needs.  By  lengthening  the  names  of  socie- 
ties (as  seems  to  have  already  begun)  the  total  can 
be  made  much  larger,  Since  the  time  when  Hoop- 
er's calculations  were  made,  two  letters  have  been 
added  to  the  alphabet.  When  the  number  of  societies 
reaches  about  the  twelve  million  mark,  it  would  be 
well  to  agitate  for  a  further  extension  of  the  alpha- 
bet. With  these  possibilities  one  may  be  assured,  on 
the  authority  of  exact  science,  that  there  is  no  cause 
for  immediate  alarm.  The  author  hastens  to  allay 
the  apprehensions  of  prospective  organizers. 

Tables.  Many  computations  would  not  be  possible 
without  the  aid  of  tables.  Some  of  them  are  monu- 
ments to  the  patient  application  of  their  makers.  Once 
made,  they  are  a  permanent  possession.  The  time 
saved  to  the  computer  who  uses  the  table  is  the  one 
item  taken  into  account  in  judging  of  the  value  of  a 
table.  It  is  difficult  to  appreciate  the  variety  and 
extent  of  the  work  that  has  been  done  in  constructing 
tables.  For  this  purpose  an  examination  of  Professor 
Glaisher's  article  "Tables"  in  the  Encyclopedia  Bri- 
tannic a  is  instructive. 

Anything  that  facilitates  the  use  of  a  book  of  tables 
is  important.  Spacing,  marginal  tabs  (in-cuts),  pro- 
jecting tabs — all  such  devices  economize  a  little  time 
at  each  handling  of  the  book ;  and  in  the  aggregate 
this  economy  is  no  trifle.  Among  American  collections 
of  tables  for  use  in  elementary  mathematics  the  best 
example   of  convenience   of   arrangement   for  ready 


40       A  SCRAP-BOOK  OF  ELEMENTARY   MATHEMATICS. 

reference  is  doubtless  Taylor's  Five-place  Logarith- 
mic and  Trigonometric  Tables  (1905).  Dietrichkeit's 
Siebenstellige  Logarithmen  und  Antilo  garithmen 
(1903)  is  a  model  of  convenience. 

When  logarithms  to  many  places  are  needed,  they 
can  be  readily  calculated  by  means  of  tables  made 
for  the  purpose,  such  as  Gray's  for  carrying  them  to 
24  places  (London,  1876). 

Factor  tables  have  been  printed  which  enable  one 
to  resolve  into,  prime  factors  any  composite  number 
as  far  as  the  10th  million.  They  were  computed  by 
different  calculators.  "Prof.  D.  N.  Lehmer,  of  the 
University  of  California,  is  now  at  work  on  factor 
tables  which  will  extend  to  the  12th  million.  When 
completed  they  will  be  published  by  the  Carnegie  In- 
stitution, Washington,  D.  C.  According  to  Petzval, 
tables  giving  the  smallest  prime  factors  of  numbers 
as  far  as  100,000,000  have  been  calculated  by  Kulik, 
but  have  remained  in  manuscript  in  the  possession  of 
the  Vienna  Academy.  .  .Lebesgue's  Table  des  Diviseurs 
des  N ombres  goes  as  far  as  115500  and  is  very  com- 
pact, occupying  only  20  pages."     (Escott.) 

Some  long  numbers.  The  computation  of  the  value 
of  w  to  707  decimal  places  by  Shanks*  and.  of  e  to 
346  places  by  Boorman,f  are  famous  feats  of  calcu- 
lation. 

"Paradoxes  of  calculation  sometimes  appear  as  il- 
lustrations of  the  value  of  a  new  method.  In  1863, 
Mr.  G.  Suffield,  M.A.,  and  Mr.  J.  R.  Lunn,  M.A.,  of 
Clare  College  and  of  St.  John's  College,  Cambridge, 
published  the  whole  quotient  of  10000.  .  .  divided  by 
7699,  throughout  the  whole  of  one  of  the  recurring 

*  See  page  124. 

f  Mathematical  Magazine,  1:204. 


MISCELLANEOUS    NOTES   ON    NUMBER.  41 

periods,  having  7698  digits.  This  was  done  in  illus- 
tration of  Mr.  Suffield's  method  of  synthetic  divi- 
sion."* 

Exceptions  have  been  found  to  Fermat's  theorem 
on  binary  powers-  (which  he  was  careful  to  say  he 
had  not  proved).  The  theorem  is,  that  all  numbers 
of  the  form  2zn  +  1  are  prime.  Euler  showed,  in 
1732,  that  if  »  =  5,  the  formula  gives  4,294,967,297, 
which  =  641  x  6,700,417.  "During  the  last  thirty  years 
it  has  been  shown  that  the  resulting  numbers  are  com- 
posite when  n-6,  9,  11,  12,  18,  23,  36,  and  38;  the 
two  last  numbers  contain  many  thousands  of  millions 
of  digits."f  To  these  values  of  n  for  which  2?n  +  1 
is  composite,  must  now  be  added  the  value  n  =  73. 
"Dr.  J.  C.  Morehead  has  proved  this  year  [1907]  that 
this  number  is  divisible  by  the  prime  number  275  •  5  +  1. 
This  last  number  contains  24  digits  and  is  probably 
the  largest  prime  number  discovered  up  to  the  pres- 
ent.":]: If  the  number  2273  +  1  itself  were  written  in 
the  ordinary  notation  without  exponents,  and  if  it 
were  desired  to  print  the  number  in  figures  the  size 
of  those  on  this  page,  how  many  volumes  like  this 
would  be  required?  They  would  make  a  library  many 
millions  of  times  as  large  as  the  Library  of  Congress. 

Hozv  may  a  particular  number  arise ?  (1)  From 
purely  mathematical  analysis — in  the  investigation  of 
the  properties  of  numbers,  as  in  the  illustrations 
just  given,  in  the   investigation  of  the  properties  of 

*  De  Morgan,  p.  292.  "Suffield's  'new'  method  was  discov- 
ered by  Fourier  in  the  early  part  of  the  century  and  has  been 
rediscovered  many  times  since.  It  was  published,  apparently 
as  a  new  discovery,  a  few  years  ago  in  the  Mathematical 
Gazette  r     (Escott.) 

i  Ball,  Recreations,  p.  37. 

%  Mr.  Escott. 


42       A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

some  ideally  constructed  magnitude,  as  the  ratio  of 
the  diagonal  to  the  side  of  a  square,  or  in  any  inves- 
tigation involving  only  mathematical  elements;  (2) 
from  measurement  of  actual  magnitude,  time  etc. : 
(3)  by  arbitrary  invention,  as  when  a  text-book  writer 
or  a  teacher  makes  examples ;  or  (4)  by  combinations 
of  these. 

Those  of  class  (3)  are  generally  used  to  develop 
skill  in  the  manipulation  of  numbers  from  classes  (1) 
and  (2). 

Numbers  from  source  (2),  measurement,  are  the 
subject  of  the  next  section. 


NUMBERS  ARISING  FROM  MEASUREMENT. 

There  is  no  such  thing  as  an  exact  measurement  of 
distance,  capacity,  mass,  time,  or  any  such  quantity. 
It  is  only  a  question  of  degree  of  accuracy. 

"The  best  time-pieces  can  be  trusted  to  measure  a 
week  within  one  part  in  756,000. "*  The  equations  of 
standards  on  page  155  show  the  degree  of  accuracy 
attained  in  two  instances  by  the  International  Bureau 
of  Weights  and  Measures.  In  the  measure  of  length 
(the  distance  between  two  lines  on  a  bar  of  platinum- 
iridium)  the  range  of  error  is  shown  to  be  0.2  in  a 
million,  or  one  in  five  million.  In  the  measure  of  mass 
it  is  one  in  five  hundred  million.  But  these  are  meas- 
urements famous  for  their  precision,  made  in  cases 
in  which  accuracy  is  of  prime  importance,  and  the 
comparisons  effected  under  the  most  favorable  con- 
ditions. No  such  accuracy  is  attained  in  most  work. 
In  a  certain  technical  school,  two-tenths  of  a  per  cent 
is  held  to  be  fair  tolerance  of  error  for.  "exact  work" 
in  chemical  analysis.  The  accuracy  in  measurement 
attained  by  ordinary  artisans  in  their  work  is  of  a 
.somewhat  lower  degree. 

Now  in  a  number  expressing  measurement  the  num- 
ber of  significant  figures  indicates  the  degree  of  ac- 
curacy. Hence  the  number  of  significant  digits  is 
limited.     If  any  one  were  to  assert  that  the  distance 

*  Prof.  William  Harkness,  "Art  of  Weighing  and  Measur- 
ing," Smithsonian  Report  for  1888,  p.  616. 

43 


44       A  SCRAP-BOOK  OF  ELEMENTARY   MATHEMATICS. 

of  Neptune  from  the  sun  is  2,788,820,653  miles,  the 
statement  would  be  immediately  rejected.  A  distance 
of  billions  of  miles  can  not,  by  any  means  now  known, 
be  measured  to  the  mile.  We  should  be  sure  that  the 
last  four  or  five  figures  must  be  unknown  and  that 
this  number  is  not  to  be  taken  seriously.  What  as- 
tronomers do  state  is  that  the  distance  is  2,788,800,000 
miles. 

The  metrology  of  the  future  will  doubtless  be  able 
to  extend  gradually  the  limits  of  precision,  and  there- 
fore to  expand  the  significant  parts  of  numbers.  But 
the  principle  will  always  hold. 

The  numbers  arising  from  the  measurements  of 
daily  life  have  but  few  significant  figures. 

The  following  paragraph  is  another  illustration  of 
the  principle. 

Decimals  as  indexes  of  degree  of  accuracy  of  meas- 
ure. The  child  is  taught  that  .42  =  .420  =  .4200.  True; 
but  the  scientist  who  reports  that  a  certain  distance  is 
.42  cm,  and  the  scientist  who  reports  it  as  .420  cm, 
wish  to  convey,  and  do  convey,  to  their  readers  differ- 
ent impressions.  From  the  first  we  understand  that 
the  distance  is  .42  cm  correct  to  the  nearest  hun- 
dredth of  a  cm;  that  is,  it  is  more  than  .415  cm  and 
less  than  .425  cm.  From  the  second  we  learn  that  it 
is  .420  cm  to  the  nearest  thousandth ;  that  is,  more 
than  .4195  and  less  than  .4205'.  Compare  the  decimals, 
including  0.00100,  in  the  equation  of  the  U.  S.  standard 
meter,  p.  155. 

Exact  measurement  is  an  ideal.  It  is  the  limit 
which  an  ever  improving  metrology  is  approaching 
forever  nearer.  The  question  always  is  of  degree  of 
accuracy  of  measure.     And  this  question  is  answered 


NUMBERS    ARISING    FROM    MEASUREMENT.  45 

by  the  number  of  decimal  places  in  which  the  result  is 
expressed. 

Some  applications.  The  foregoing  principle  ex- 
plains why  for  very  large  and  very  small  numbers  the 
index  notation  is  sufficient ;  in  which  it  is  said,  for 
example,  that  a  certain  star  is  5  x  1013  miles  from  the 
earth.  This  is  easier  to  write  than  5  followed  by  13 
ciphers,  and  there  is  no  need  to  enumerate  and  read 
such  a  number.  Similarly  10  with  a  negative  exponent 
serves  to  write  such  a  decimal  fraction  as  is  used  to 
express  the  length  of  a  wave  of  light  or  any  of  the 
minute   measurements   of  microscopy. 

The  principle  explains  also  why  a  table  of  loga- 
rithms for  ordinary  use  need  not  tabulate  numbers  be- 
yond four  or  five  places  (four  or  five  places  in  the 
"arguments/'  to  use  the  technical  term  of  table  makers  ; 
only  the  logarithms  of  numbers  to  10,000,  or  100,000, 
to  use  the  common  phraseology).  Interpolation  ex- 
tends them  to  one  more  place  with  fair  accuracy,  and 
for  ordinary  computation  one  rarely  needs  the  loga- 
rithm of  a  number  of  more  than  five  significant  digits. 

It  explains  also  why  a  method  of  approximation  in 
multiplication  is  so  desirable.  If  any  of  the  data  are 
furnished  by  measurement,  the  result  can  be  only 
approximate  at  best.  Example  VII  on  page  64,  ex- 
plained on  page  62,  is  a  case  in  point.  To  compute 
that  product  to  six  decimal  places  would  waste  time. 
Worse  than  that ;  to  show  such  a  result  would  pretend 
to  an  accuracy  not  attained,  by  conveying  the  impres- 
sion that  the  circumference  is  known  to  six  decimal 
places  when  in  fact  it  is  known  to  but  two  decimal 
places/1' 

*  Even  the  second  decimal  place  is  in  doubt,  as  may  be 
seen  by  taking  for  multiplicand  first  74.276,  then  74.284. 


46       A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

In  a  certain  village  the  tax  rate,  found  by  dividing 
the  total  appropriation  for  the  year  by  the  total  as- 
sessed valuation,  was  .01981  for  the  year  1906.  As 
always  (unless  the  divisor  contains  no  prime  factor 
but  2  and  5)  the  quotient  is  an  interminate  decimal. 
To  how  many  places  should  the  decimal  be  carried? 
Theoretically  it  should  be  carried  far  enough  to  give 
a  product  "correct  to  cents"  when  used  to  compute 
the  tax  of  the  highest  taxpayer.  In  this  case  the 
decimal  is  accurate  enough  for  all  assessments  not 
exceeding  $1000.  As  a  matter  of  fact,  there  were 
several  in  excess  of  this  amount. 

For  an  understanding  of  the  common  applications 
of  arithmetic  it  is  important  that  the  learner  appre- 
ciate the  elementary  considerations  of  the  theory  of 
error ;  at  least  that  he  habitually  ask  himself,  "To  how 
many  places  may  my  result  be  regarded  as  accurate  ?" 


COMPOUND  INTEREST. 

The  enormous  results  obtained  by  computing  com- 
pound interest — as  well  as  the  wide  divergence  be- 
tween these  or  any  results  obtained  from  a  geometric 
progression  of  many  terms  and  the  results  found  in 
actual  life — may  be  seen  from  the  following  "ex- 
amples" : 

At  3%  (the  prevailing  rate  at  present  in  savings 
banks)  $1  put  at  interest  at  the  beginning  of  the 
Christian  era  to  be  compounded  annually  would  now 
amount  to  $(1.03)1906,  which  by  the  use  of  logarithms 
is  found  to  be,  in  round  numbers,  $3,000,000,000,000,- 
000,000,000,000.  The  amount  of  $1  for  the  same 
time  and  rate  but  at  simple  interest  would  be  only 
$58.18. 

If  the  Indians  hadrit  spent  the  $24.  In  1626  Peter 
Minuit,  first  governor  of  New  Netherland,  purchased 
Manhattan  Island  from  the  Indians  for  about  $24. 
The  rate  of  interest  on  money  is  higher  in  new  coun- 
tries, and  gradually  decreases  as  wealth  accumulates. 
Within  the  present  generation  the  legal  rate  in  the 
state  has  fallen  from  7%  to  6%.  Assume  for  sim- 
plicity a  uniform  rate  of  7%  from  1626  to  the  present, 
and  suppose  that  the  Indians  had  put  their  $24  at  in- 
terest at  that  rate  (banking  facilities  in  New  York 
being  always  taken  for  granted ! )   and  had  added  the 

47 


48       A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

interest  to  the  principal  yearly.  What  would  be  the 
amount  now,  after  280  years?  24xl.07280=  more 
than  4,042,000,000. 

The  latest  tax  assessment  available  at  the  time 
of  writing  gives  the  realty  for  the  borough  of  Man- 
hattan as  $3,820,754,181.  This  is  estimated  to  be 
78%  of  the  actual  value,  making  the  actual  value  a 
little  more  than  $4,898,400,000. 

The  amount  of  the  Indians'  money  would  therefore 
be  more  than  the  present  assessed  valuation  but  less 
than  the  actual  valuation.  The  Indians  could  have 
bought  back  most  of  the  property  now,  with  improve- 
ments;  from  which  one  might  point  the  moral  of 
saving  money  and  putting  it  at  interest!  The  rise 
in  the  value  of  the  real  estate  of  Manhattan,  phenom- 
enal as  it  is,  has  but  little  more  than  kept  pace  with 
the  growth  of  money  at  7%  compound  interest.  But 
New  York  realty  values  are  now  growing  more  rap- 
idly: the  Indians  would  better  purchase  soon! 


DECIMAL   SEPARATRIXES. 

The  term  separafrix  in  the  sense  of  a  mark  between 
the  integral  and  fractional  parts  of  a  number  written 
decimally,  was  used  by  Oughtred  in  1631.  He  used 
the  mark  L  for  the  purpose.  Stevin  had  used  a 
figure  in  a  circle  over  or  under  each  decimal  place  to 
indicate  the  order  of  that  decimal  place.  Of  the  vari- 
ous other  separatrixes  that  have  been  used,  four  are 
in  common  use  to-day,  if  (2)  and  (3)  below  may  be 
counted  separately: 

1.  A  vertical  line:  e.g.,  that  separating  cents  from 
dollars  in  ledgers,  bills  etc.  As  a  temporary  sepa- 
ratrix  the  line  appears  in  a  work  by  Richard  Witt, 
1613.  Napier  used  the  line  in  his  Rabdologia,  1617. 
This  is  a  very  common  separatrix  in  every  civilized 
country  to-day. 

2.  The  period.  Fink,  citing  Cantor,  says  that  the 
decimal  point  is  found  in  the  trigonometric  tables  of 
Pitiscus  (in  Germany)  1612.  Napier,  in  the  Rabdo- 
logia, speaks  of  using  the  period  or  comma.  His 
usage,  however,  is  mostly  of  a  notation  now  obsolete 
(but  he  uses  the  comma  at  least  once).  The  period 
has  always  been  the  prevailing  form  of  the  decimal 
point  in  America. 

3.  The  Greek  colon  (dot  above  the  line).  Newton 
advocated  placing  the  point  in  this  position  "to  pre- 
vent it  from  being  confounded  with  the  period  used 


49 


50      A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

as  a  mark  of  punctuation1'  (Brooks).  It  is  commonly 
so  written  in  England  now. 

4.  The  comma.  The  first  known  instance  of  its  use 
as  decimal  separatrix  is  said  to  be  in  the  Italian  trigo- 
nometry of  Pitiscus,  1608.  Perhaps  next  by  Kepler, 
1616,  from  which  may  be  dated  the  German  use  of  it. 
Briggs  used  it  in  his  table  of  logarithms  in  1624r  and 
early  English  writers  generally  employed  the  comma. 
English  usage  changed  to  the  Greek  colon;  but  the 
comma  is  the  customary  form  of  the  decimal  point  on 
the  continent  of  Europe. 

The  usage  as  to  decimal  point  is  not  absolutely 
uniform  in  any  of  the  countries  named ;  but,  in  gen- 
eral, one  expects  to  see  1 25/100  written  decimally  in 
the  form  of  1 .  25  in  America,  1-25  in  England,  and 
1,25  in  Germany,  France  or  Italy. 

A  mere  space  to  indicate  the  separation  may  also 
be  mentioned  as  common  in  print. 

The  vertical  line  (for  a  column  of  decimals)  and  the 
space  should  doubtless  persist,  and  one  form  of  the 
"point."  Prof.  G.  A.  Miller,  of  the  University  of 
Illinois,-  who  argues  for  the  comma  as  being  the  symbol 
used  by  much  the  largest  number  of  mathematicians, 
remarks  :*  "As  mathematics  is  pre-eminently  cosmo- 
politan and  eternal  it  is  very  important  that  its  symbols 
should  be  world  symbols.  All  national  distinctions 
along  this  line  should  be  obliterated  as  rapidly  as 
possible." 

*  "On  Some  of  the  Symbols  of  Elementary  Mathematics," 
School  Science  and  Mathematics,  May,  1907. 

Where  the  decimal  point  is  a  comma  the  separation  of  long 
numbers  into  periods  of  three  (or  six)  figures  for  convenience 
of  reading  is  effected  by  spacing. 


PRESENT  TRENDS  IN  ARITHMETIC. 

"History  is  past  politics,  and  politics  is  present  his- 
tory/' Such  is  the  apothegm  of  the  famous  historian 
Freeman.  In  the  case  of  a  science  and  an  art,  like 
arithmetic  or  the  teaching  of  arithmetic,  history  is 
past  method,  and  method  is  present  history.  The  fact 
that  our  generation  is  helping  to  make  the  history  of 
arithmetic  and  of  the  teaching  of  arithmetic — as  it  is 
also  making  history  in  other  matters  that  attract  more 
public  attention — is  the  reason  for  considering  now 
some  of  the  present  trends  in  arithmetic.  A  present 
trend  is  a  pointer  pointing  from  what  has  been  to 
what  is  to  be,  since  the  science  is  a  continuum.  Lord 
Bolingbroke  said  that  we  study  history  to  know  how 
to  act  in  the  future,  to  make  the  most  of  the  future. 
That  is  why  we  study  history  in  the  making,  or  pres- 
ent trends,  in  so  far  as  it  is  possible  for  us,  living  in 
the  midst,  to  see  those  trends. 

Very  noticeable  among  them  is  the  gradual  deci- 
malization of  arithmetic.  Counting  by  10  is  prehistoric 
in  nearly  all  parts  of  the  world,  10  fingers  being  the 
evident  explanation.  If  we  had  been  present  at  the 
beginning  of  arithmetical  history,  we  might  have  given 
the  primitive  race  valuable  advice  as  to  the  choice  of 
a  radix  of  notation !  It  would  then  have  been  oppor- 
tune to  call  attention  to  the  advantage  of  12  over  10 
arising  from  the  greater  factorability  of   12.     Or  if 

5i 


52       A  SCRAP-BOOK  OF  ELEMENTARY   MATHEMATICS. 

the  pioneers  of  arithmetic  had  been  like  the  Gath 
giant  of  2  Sam.  21 :  20,  with  six  ringers  on  each  hand, 
they  would  doubtless  have  used  12  as  a  radix.  Lack- 
ing such  counsel,  and  being  equipped  by  nature  with 
only  10  fingers  to  use  as  counters,  they  started  arith- 
metic on  a  decimal  basis.  History  since  has  been  a 
steady  progress  in  the  direction  thus  chosen  (except 
in  details  like  the  table  of  time,  where  the  incommen- 
surable ratio  between  the  units  fixed  by  nature  defied 
even  the  French  Revolution). 

The  Arabic  notation  "was  brought  to  perfection  in 
the  fifth  or  sixth  century,"*  but  did  not  become  com- 
mon in  Europe  till  the  sixteenth  century.  It .  is  not 
quite  universal  yet,  the  Roman  numerals  being  still 
used  on  the  dials  of  timepieces,  in  the  titles  of  sov- 
ereigns, the  numbers  of  book  chapters  and  subdivi- 
sions, and,  in  general,  where  an  archaic  effect  is 
sought.  But  the  Arabic  numerals  are  so  much  more 
convenient  that  they  are  superseding  the  Roman  in 
these  places.  The  change  has  been  noticeable  even 
in  the  last  ten  or  fifteen  years. 

The  extension  of  the  Arabic  system  to  include  frac- 
tions was  made  in  the  latter  part  of  the  sixteenth 
century.  But  notwithstanding  the  superior  conve- 
nience of  decimal  fractions,  they  spread  but  slowly  ; 
and  it  is  only  in  comparatively  recent  times  that  they 
may  be  said  to  be  more  common  than  "common  frac- 
tions." 

The  next  step  was  logarithms — a  step  taken  in  1614. 
Within  the  next  ten  years  they  were  accomodated  to 
what  we  should  call  "the  base"  10. 

The  dawn  of  the  nineteenth  century  found  decimal 
coinage  well  started  in  the  United  States,  and  a  gen- 

*  Cajori,  Hist,  of  Eicm.  Math.,  p.   154. 


PRESENT  TRENDS  IN   ARITHMETIC.  53 

eral  movement  toward  decimalization  under  way  in 
France  contemporaneous  with  the  political  revolution. 
The  subsequent  spread  of  the  metric  system  over  most 
of  the  continent  of  Europe  and  over  many  other  parts 
of  the  world  has  been  the  means  of  teaching  decimal 
fractions. 

The  movement  is  still  on.  The  value  and  impor- 
tance of  decimals  are  now  recognized  more  every  year. 
And  much  remains  to  be  decimalized.  In  stock  quo- 
tations, fractions  are  not  yet  expressed  decimally. 
Three  great  nations  have  still  to  adopt  decimal,  weights 
and  measures  in  popular  use,  and  England  has  still  to 
adopt  decimal  coinage.  The  history  of  arithmetic  has 
been,  in  large  part,  a  slow  but  well-marked  growth  of 
the  decimal  idea. 

Those  who  are  working  for  world  -  wide  decimal 
coinage,  weights  and  measures — as  a  time-saver  in 
school-room,  counting  house  and  work-shop — as  a 
boon  that  we  owe  to  posterity  as  well  as  to  ourselves 
— may  learn  from  such  a  historical  survey  both  cau- 
tion and  courage.  Caution  not  to  expect  a  sudden 
change.  Multitudes  move  slowly  in  matters  requiring 
a  mental  readjustment.  The  present  reform  move- 
ments —  for  decimal  weights  and  measures  in  the 
United  States,  and  decimal  weights,  measures  and  coins 
in  Great  Britain — are  making  more  rapid  progress  than 
the  Arabic  numerals  or  decimal  fractions  made ;  and 
the  opponents  of  the  present  reform  are  not  so  numer- 
ous or  so  prejudiced  as  were  their  prototypes  who 
opposed  the  Arabic  notation  in  the  Middle  Ages  and 
later.  Caution  also  against  impatience  with  a  con- 
servatism whose  arguments  are  drawn  from  the  tem- 
porary inconvenience  of  making  any  change.  Cour- 
age to  work  and  wait — in  line  with  progress. 


54       A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

In  using  fractions,  the  Egyptians  and  Greeks  kept 
the  numerators  constant  and  operated  with  the  denom- 
inators. The  Romans  and  Babylonians  preferred  a 
constant  denominator,  and  performed  operations  on 
the  numerator.  The  Romans  reduced  their  fractions 
to  the  common  denominator  12,  the  Babylonians  to 
60ths.  We  also  reduce  our  fractions  to  a  common 
denominator ;  but  we  choose  100.  One  of  the  most 
characteristic  trends  of  modern  arithmetic  is  the  rapid 
growth  in  the  use  of  percentage — another  development 
of  the  decimal  idea.  The  broker  and  the  biologist,  the 
statistician  and  the  salesman,  the  manufacturer  and  the 
mathematician  alike  express  results  in  per  cents. 

These  and  other  changes  in  the  methods  of  com- 
puters have  brought  about,  though  tardily,  correspond- 
ing changes  in  the  subject  matter  of  arithmetic  as 
taught  in  the  schools.  Scholastic  puzzles  are  giving 
place  to  problems  drawn  from  the  life  of  to-day. 

Perhaps  one  may  venture  the  opinion  that,  in  order 
to  merit  a  place  in  the  arithmetic  curriculum,  a  topic 
must  be  useful  either  (1)  in  commerce  or  (2)  in  in- 
dustry or  (3)  in  science.  Under  (3)  may  be  included, 
conceivably,  a  topic  whose  sole,  or  chief,  use  is  in  later 
mathematical  work.  At  least  two  other  reasons  have 
been  given  for  retaining  a  subject:  (4)  It  is  required 
for  examination.  But  it  will  be  found  that  subjects 
not  clearly  justified  on  one  of  the  grounds  above'men- 
tioned  are  rarely  required  by  examining  bodies  of  this 
generation  ;  and  such  subjects,  if  pointed  out,  would 
doubtless  be  withdrawn  from  any  syllabus.  (5)  It 
gives  superior  mental  training.  But  on  closer  scrutiny 
this  argument  becomes  somewhat  evanescent.  A  sur- 
vey of  results  in  that  branch  of  educational  psychology 


PRESENT  TRENDS  IN  ARITHMETIC.  55 

which  treats  of  the  coefficient  of  correlation  between 
a  pupil's  attainments  in  various  activities,  weakens 
one's  faith  in  our  ability  to  give  a  certain  amount  of 
general  discipline  by  a  certain  amount  of  special  train- 
ing. Moreover,  that  discipline  can  be  as  well  acquired 
by  the  study  of  subjects  that  serve  a  direct,  useful 
purpose.  We  may,  then,  limit  our  criteria  to  these: 
utility  for  business  or  industrial  pursuits,  and  utility 
for  work  in  science. 

Applying  these  tests  to  the  topics  contained  in  the 
schoolbooks  of  a  generation  ago,  we  see  that  many 
of  them  are  not  worthy  of  a  place  in  the  crowded  cur- 
riculum of  our  generation.  Turning  to  the  schools, 
we  find  that  many  of  these  topics  have,  in  fact,  been 
dropped.  Others  are  receiving  less  attention  each 
year.  Among  such  may  be  mentioned :  "true"  dis- 
count, partnership  involving  time,  and  equation  of 
payments  (all  three  giving,  besides,  a  false  idea  of 
business),  and  Troy  and  apothecaries  weight,  cube 
root  (except  for  certain  purposes  with  advanced  pu- 
pils) and  compound  proportion. 

At  the  same  time,  other  topics  in  the  arithmetic 
course  are  of  increasing  importance ;  notably  those 
involving  percentage  and  other  decimal  operations, 
and  those  relating  to  stock  companies  and  other  devel- 
opments of  modern  economic  activity. 

School  life  is  adjusting  itself  to  present  social  con- 
ditions, not  only  in  the  topics  taught,  but  in  the  prob- 
lems used  and  the  way  in  which  the  topics  are  treated. 
Good  books  no  longer  set  problems  in  stocks  involv- 
ing the  purchase  of  a  fractional  number  of  shares ! 

As  Agesilaus,  king  of  Sparta,  said,  "Let  boys  study 
what  will  be  useful  to  men." 


56       A  SCRAP-ROOK  OF  ELEMENTARY    MATHEMATICS. 

The  Greeks  studied  d/oify^™*^,  or  theory  of  numbers, 
and  XoyicrTiKr),  or  practical  calculation.  Hence  the  mod- 
ern definition  of  arithmetic,  "the  science  of  numbers 
and  the  art  of  computation."  As  Prof.  David  Eugene 
Smith  points  out  (in  his  Teaching  of  Elementary 
Mathematics)  "the  modern  arithmetic  of  the  schools 
includes  much  besides  this."  It  includes  the  introduc- 
tion of  the  pupil  to  the  commercial,  industrial  and 
scientific  life  of  to-day  on  the  quantitive  side. 

Characteristic  of  our  time  is  the  extensive  use  of 
arithmetical  machines  (such  as  adding  machines  and 
instruments  from  which  per  cents  may  be  read)  and 
of  tables  (of  square  roots  for  certain  scientific  work, 
interest  tables  for  banks,  etc.).  The  initial  invention 
of  such  appliances  is  not  recent ;  it  is  their  variety, 
adaptability  and  rapidly  extending  usefulness  that  may 
be  classed  as  a  present  phenomenon. 

They  have  not,  however,  eliminated  the  necessity 
for  training  good  reckoners.  They  may  have  nar- 
rowed the  field  somewhat,  but  in  that  remaining  part 
which  is  both  practical  and  necessary  they  have  set 
the  standard  of  attainment  higher.  Indeed,  an  im- 
portant feature  of  the  present  situation  is  the  insistent 
demand  of  business  men  that  the  schools  turn  out  bet- 
ter computers.  There  must  soon  come,  in  school,  a 
stronger  emphasis  on  accuracy  and  rapiditv  in  the 
four  fundamental  operations- 
Emphasis  on  accuracy  and  rapidity  in  calculation 
leads  to  the  use  of  "examples"  involving  abstract 
numbers.  Emphasis  on  the  business  applications  alone 
leads  to  the  almost  exclusive  use  of  "problems' '  in 
which  the  compilative  is  but  an  incidental  feature. 
Both  are  necessarv.     It  has  been  well  said  that  exam- 


PRESENT  TRENDS  IN   ARITHMETIC.  57 

pies  are  to  the  arithmetic  pupil  what  exercises  are  to 
the  learner  on  the  piano,  while  problems  are  to  the 
former  what  tunes  are  to  the  latter.  Without  exer- 
cises, no  skill ;  with  exercises  alone,  no  accomplish- 
ment. The  exercises  are  for  the  technique  of  the  art. 
The  teacher  can  not  afford  to  neglect  either. 

The  last  century  or  more  has  been  the  age  of  special 
methods  in  teaching.  One  has  succeeded  another  in 
popular  favor.  Each  has  taught  us  an  important 
lesson — something  that  will  be  a  permanent  acquisition 
to  the  pedagogy  of  the  science.  Few  things  are  more 
interesting  to  the  student  of  the  history  of  arithmetic 
methods  than  to  trace  each  school-room  practice  of 
to-day  to  its  origin  in  some  worthy  contributor  to  the 
science  (e.g.,  in  the  primary  grades,  the  use  of  blocks 
to  Trapp,  1780;  the  "number  pictures"  to  Von  Busse ; 
counting  by  2's,  3's.  .  .  as  preparation  for  the  multi- 
plication tables  to  Knilling  and  Tanck ;  etc.).  More 
recently  several  famous  methods  have  appeared  which 
are  still  advocated.  But  the  present  trend  is  toward 
a  choosing  of  the  best  from  each — an  eclectic  method. 

Most  questions  of  method  have  never  been  ade- 
quately tested.  It  is,  for  instance,  asserted  by  some 
and  denied  by  others  that  pupils  would  know  as  much 
arithmetic  at  the  end  of  the  8th  school  year  if  they 
were  to  begin  arithmetic  with  the  5th  or  even  later. 
History  may  well  lead  us  to  doubt  the  proposition ; 
but  who  can  tell?  The  greatest  desideratum  in  all 
arithmetic  teaching  to-day  is  the  thorough  study  of 
the  subject  by  the  scientific  methods  employed  in 
educational  psychology.  Some  one  with  facilities  for 
doing  this  service  for  arithmetic  could  be  a  benefactor 


58      A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

indeed.  Questions  that  are  matters  for  accurate  test 
and  measurement  should  not  always  remain  questions. 
Meantime,  empiricism  is  unavoidable. 

To  summarize  the  tendencies  noted :  the  decimali- 
zation of  arithmetic,  growth  of  percentage,  elimina- 
tion of  many  topics  from  the  school  curriculum  in 
arithmetic  with  increased  emphasis  on  others,  mod- 
ernizing the  treatment  of  remaining  topics,  demand 
for  more  accuracy  and  rapidity  in  computation,  in- 
clination toward  an  eclectic  method  in  teaching  arith- 
metic, present  empiricism  pending  scientific  investi- 
gation. This  list  is,  of  course,  far  from  exhaustive, 
but  it  is  believed  to  be  true  and  significant. 

Lacking  such  exact  information  as  that  just  asked 
for  as  the  desideratum  of  to-day,  w  e  may  make  the  best 
of  mere  observation  of  the  trends  of  our  time.  And 
as  to  the  great  movements  in  the  history  of  the  art  of 
arithmetic  itself,  the  conclusions  are  definite  and  de- 
cisive. By  orienting  ourselves,  by  studying  the  past 
and  noting  the  currents,  we  may  acquaint  ourselves 
with  the  direction  of  present  forces  and  may  take  part 
in  shaping  our  course.  Our  to-days  are  conditioned 
by  our  yesterdays,  to-morrow  by  to-day. 


MULTIPLICATION  AND  DIVISION  OF  DECI- 
MALS. 

For  the  multiplication  of  whole  numbers  the  Italians 
invented  many  methods.*  Pacioli  (1494)  gives  eight. 
Of  these,  only  one  was  in  common  use,  and  it  alone 
has  survived  in  commerce  and  the  schools.  Shown 
in  I  on  p.  64.  It  was  called  bericiiocolo  (honey  cake  or 
ginger  bread)  by  the  Florentines,  and  scacchiera  (chess 
or  checker  board)  by  the  Venetians.  The  little  squares 
in  the  partial  products  fell  into  disuse  (and  with  them 
the  names  which  they  made  appropriate)  leaving  the 
familiar  form  II  on  p.  64.  The  Treviso  arithmetic 
(1478),  the  first  arithmetic  printed,  contains  a  long 
example  in  multiplication,  which  appears  about  as  it 
would  appear  on  the  blackboard  of  an  American  school 
to-day. 

In  1585  appeared  Simon  Stevin's  immortal  La 
Disme,  only  seven  pages,  but  the  first  publication  to 
expound  decimal  fractions,  though  the  same  author 
had  used  them  in  an  interest  table  published  the  year 
before.  Ill  on  p.  64  is  from  La  Disme,  and  shows 
Stevin's  notation  (the  numbers  in  circles,  or  paren- 
theses, indicating  the  order  of  decimals,  tenths  the 
first  order  etc.)  IV  is  the  same  example  with  the 
decimals  expressed  by  the  notation  now  prevalent  in 

*  For  the  historical  facts  in  this  section  the  author  is  in- 
debted mainly  to  Professor  Cajori  and  Prof.  David  Eugene 
Smith,  the  two  leading  American  authorities  on  the  history  of 
mathematics. 

59 


6o       A  SCRAP-BOOK  OF  ELEMENTARY   MATHEMATICS. 

America.  Let  us  call  this  arrangement  of  work  Ste- 
vin's  method. 

An  arrangement  in  which  all  decimal  points  are  in 
a  vertical  column  (see  V  below)  is  said  to  have  been 
used  by  Adrian  Romain  a  quarter  of  a  century  later. 
He  may  not  have  been  the  inventor  of  this  arrange- 
ment ;  but,  for  the  sake  of  a  name,  call  it  Romain's 
method. 

Romain 's  method  is  advocated  in  a  few  of  the  best 
recent  advanced  arithmetics,  but  Stevin's  is  still  vastly 
the  more  common  ;  and  these  two  are  the  only  methods 
in  use.  Romain's  has  four  slight  advantages :  ( 1 )  A 
person  setting  down  an  example  from  dictation  can 
begin  to  write  the  multiplier  as  soon  as  the  place  of  its 
decimal  point  is  seen,  while  in  Stevin's  method  he 
waits  to  hear  the  entire  multiplier  before  he  writes 
any  of  it,  in  order  to  have  its  last  (right-hand)  figure 
stand  beneath  the  last  figure  of  the  multiplicand 
(though  this  positon  may  be  regarded  as  a  non- 
essential feature  in  Stevin's  arrangement).  (2)  Ro- 
main's method  fits  more  naturally  with  the  "Austrian" 
method  of  division  (decimal  point  of  quotient  over 
that  of  dividend).  (3)  After  the  partial  products 
are  added,  it  is  not  necessary  to  count  and  point  off 
in  the  product  as  many  decimal  places  as  there  are  in 
the  multiplicand  and  multiplier  together,  since  the 
decimal  point  in  the  product  (as  well  as  in  the  partial 
products)  is  directly  beneath  that  in  the  multiplicand. 
(4)  Romain's  method  is  more  readily  adapted  to 
abridged  multiplication  where  only  approximate  re- 
sults are  required.  On  the  other  hand,  Stevin's  method 
has  one  very  decided  advantage :  the  first  figure  writ- 
ten in  each  partial  product  is  directly  beneath  its 
figure   in   the   multiplier,   so   that   it   is   not   necessary 


MULTIPLICATION    OF  DECIMALS.  6 1 

(as  it  is  in  Romaiirs)  to  determine  the  place  of  the 
decimal  point  in  a  partial  product.  So  important  is 
this,  that  Stevin's  alone  has  been  generally  taught  to 
children,  notwithstanding  the  numerous  points  in  favor 
of  Romain's. 

It  occurred  to  the  writer  recently  to  try  to  combine 
in  one  method  the  advantages  of  both  of  the  Flemish 
methods,  and  he  hit  upon  the  following  simple  rule  :* 
Write  the  units  figure  of  the  multiplier  under  the  last 
(right-hand)  figure  of  the  multiplicand,  begin  each 
partial  product  (as  in  the  familiar  method  of  Stevin) 
under  the  figure  by  which  you  are  multiplying,  and  all 
decimal  points  in  products  will  then  be  directly  be- 
neath that  in  the  multiplicand.  Decimal  points  in 
partial  products  may  be  written  or  not,  as  desired. 
The  reason  underlying  the  rule  is  apparent.  VI  shows 
the  arrangement  of  work. 

In  this  arrangement  the  placing  of  the  partial  pro- 
ducts is  automatic,  as  in  Stevin's  method,  and  the 
pointing  off  in  the  product  is  automatic,  as  in  Ro- 
main's. It  is  available  for  use  by  the  child  in  his  first 
multiplication  of  decimals  and  by  the  skilled  com- 
puter in  his  abridged  work. 

To  assist  in  keeping  like  decimal  orders  in  the  same 
column  it  is  recommended  that  the  vertical  line  shown 
in  VII  and  VIII  be  drawn  before  the  partial  prod- 
ucts are  written.  One  of  the  earliest  uses  of  the  line 
as  decimal  separatrix  is  in  an  example  in  Napier's 
Rabdologia  (1617).  He  draws  it  through  the  partial 
and    complete    products.      It    is    said   to   be   the   first 

*  Since  writing  this  the  author  has  come  upon  the  same 
method  of  multiplication  in  Lagrange's  Lectures,  delivered  in 
1795  (p-  29-30  of  the  Open  Court  Publishing  Co.'s  edition). 
One  who  invents  anything  in  elementary  mathematics  is  likely 
to  find  that  "the  ancients  have  stolen  his  ideas." 


62       A  SCRAP-BOOK  OF  ELEMENTARY   MATHEMATICS. 

example  of  abridged  multiplication.  A  circumference 
is  computed  whose  diameter  is  635. 

VII  illustrates  the  application  of  the  method  here 
advocated  to  multiplication  in  which  only  an  approxi- 
mation is  sought.  The  diameter  of  a  circle  is  found 
by  measurement  to  be  74.28  cm.  This  is  correct  to 
0.01  cm.  No  computation  can  give  the  circumference 
to  any  higher  degree  of  accuracy.  Partial  products 
are  kept  to  three  places  in  order  to  determine  the  cor- 
rect figure  for  the  second  place  in  the  complete  pro- 
duct. The  arrangement  of  work  shows  what  figures 
to  omit. 

It  should  be  remarked  that  all  three  methods  of 
multiplication  of  decimals  are  alike — and  like  the  mul- 
tiplication of  whole  numbers — in  that  one  may  multi- 
ply first  by  the  digit  of  lowest  order  in  the  multiplier 
or  by  the  digit  of  highest  order  first.  The  method  of 
multiplying  by  the  highest  order  first  was  described 
by  the  Italian  arithmeticians  as  a  dietro.  Though  it 
may  seem  to  be  working  backwards,  it  is  not  so  in 
fact ;  for  it  puts  the  more  important  before  the  less, 
and  has  practical  advantage  in  abridged  multiplica- 
tion, like  that  shown  in  VII.  But  that  question  is 
distinct  from  the  one  under  consideration. 

Stevin  writes  the  last  figure  of  the  multiplier  under 
the  last  figure  of  the  multiplicand ;  Romain  writes 
units  under  units ;  the  method  here  proposed  writes 
units  under  last.  In  whole  numbers,  units  figure  is 
the  last. 

Applied  to  the  ordinary  multiplication  of  decimals, 
as  in  VI  or  VIII,  the  method  here  proposed  seems 
to  be  well  adapted  to  schoolroom  use,  possessing  all 
the  simplicity  of  Stevin's.  Methods  classes  in  this 
normal   school  to   whom  the  method   was   presented, 


MULTIPLICATION   OF  DECIMALS.  63 

immediately  preferred  it,  and  a  grade  in  the  training- 
school  used  it  readily.  Of  course  this  proves  nothing, 
for  every  method  is  a  success  in  the  hands  of  its  ad- 
vocates. The  changes  here  set  forth  are,  however, 
not  advocated ;  they  are  merely  proposed  as  a  possi- 
bility. 

The  analogous  method  for  division  of  decimals 
possesses  analogous  advantages.  It  avoids  the  ne- 
cessity of  multiplying  the  divisor  and  dividend  by  such 
a  power  of  10  as  will  make  the  divisor  integral  (as  in 
the  method  now  perhaps  most  in  favor)  and  the  ne- 
cessity of  counting  to  point  of!  in  the  quotient  a  num- 
ber of  decimal  places  equal  to  the  number  in  the 
dividend  minus  that  in  the  divisor  (as  in  the  older 
method  still  common).  Like  the  latter,  it  begins  the 
division  at  once  ;  and  like  the  former,  its  pointing  off 
is  automatic.  IX  shows  the  arrangement.  The  figure 
under  the  last  figure  of  the  divisor  is  units  figure  of  the 
quotient.  This  determines  the  place  of  the  decimal 
point.  That  part  of  the  quotient  which  projects  beyond 
the  divisor,  is  fractional. 

If  the  order  of  multiplication  used  has  been  a  dietro, 
as  in  VIII,  the  division  in  IX  is  readily  seen  to  be 
the  inverse  operation.  The  partial  products  appear  in 
the  same  order  as  partial  dividends. 

Like  each  of  the  methods  in  use,  it  may  be  abbre- 
viated by  writing  only  the  remainders  below  the  divi- 
dend.    Shown  in  X. 

If  the  "little  castle"  method  of  multiplication  of 
whole  numbers,  with  multiplier  above  multiplicand, 
had  prevailed,  instead  of  the  "chess  board,"  in  the 
fifteenth  century,  the  arrangement  now  proposed  for 
the  multiplication  and  division  of  decimals  would  have 
afforded  slightly  greater  advantage. 


64       A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 


3  2  5  7 
8  9  4  6 

l 

9 

542J 

1 

3 

0 

2 

8 

2 

9 

3 

1 

3 

2 

<» 

0 

3 

6 

2  9  13  7  12  2 


II 


3  2  5 

7 

8  9  4  6 

1 

9  5  4 

2 

1  3 

0  2  8 

2  9  3 

1  3 

2  6  0  5 

6 

2  9  13  7  12  2 


III 

IV 

(0)(l)(2) 

3  2  5  7 

.    3  2.5  7 

8  9  4  6 

8  9.4  6 

19  5  4  2 

19  5  4  2 

13  0  2  8 

13  0  2  8 

2  9  3  13 

2  9  3  13 

2  6  0  5  6 

2  6  0  5  6 

2  9  13  7  12  2     . 

2  9  1  3.7  1  2  2 

(0)  (I)  (2)  (3)  (4) 

V 

VI 

3  2.5  7 

3  2.5  7 

8  9.4  6 

8  9.4  6 

1.9  5  4  2 

1.9  5  4  2 

1  3.0  2  8 

1  3.0  2  8 

2  9  3.1  3 

2  9  3.1  3 

2  6  0  5.6 

2  6  0  5.6 

2  9  13.7122 

2  9  1  3.7  1  2  2 

VII 

VIII 

7  4.2  8 

3  2.5  7 

3.1  4  1  6 

8  9.4  6 

2  2  2 

8  4 

7 

4  2  8 

2 

9  7  1 

7  4 

4  5 

2  3  3 

3  6 

2  6  0  5j6 

2  9  3 

1  3 

1  3 

0  2  8 

1 

9  5  4  2 

2  9  13 

7  12  2 

MULTIPLICATION   OF  DECIMALS.  65 


IX 

X 

divisor 

3  2.5  7 

3  2.5  7 

quotient 

8  9.4  6 
2  9  1  3.7  1  2  2 

8  9.4  6 

dividend 

2  9  13.7122 

2  6  0  5  6 

3  0  8  11 

3  0  8  11 

14  9  8  2 

2  9  3  13 

19  5  4  2 

14  9  8  2 

13  0  2  8 

19  5  4  2 

19  5  4  2 

ARITHMETIC   IN   THE   RENAISSANCE. 

The  invention  of  printing  was  important  for  arith- 
metic, not  only  because  it  made  books  more  accessible, 
but  also  because  it. spread  the  use  of  the  Hindu  ("Ar- 
abic") numerals  with  their  decimal  notation. 

The  oldest  text-book  on  arithmetic  to  use  these  nu- 
merals is  said  to  be  that  of  Avicenna,  an  Arabian 
physician  of  Bokhara,  about  1000  A.  D.  (firooks). 
According  to  Cardan  (sixteenth  century)  it  was  Leo- 
nardo of  Pisa  who  introduced  the  numerals  into  Eu- 
rope (by  his  Liber  Abaci,  1202).  In  England,  though 
there  is  one  instance  of  their  use  in  a  manuscript  of 
1282,"  and  another  in  1325,  their  use  is  somewhat  ex- 
ceptional even  in  the  fifteenth  century.  Then  came 
printed  books  and  a  more  general  acceptance  of  the 
decimal  notation. 

The  importance  of  this  step  can  hardly  be  over- 
estimated. Even  the  Greeks,  with  all  their  mathemat- 
ical acumen,  had  contented  themselves  with  mystic  and 
philosophic  properties  of  numbers  and  had  made  com- 
paratively little  progress  in  the  art  of  computation. 
They  lacked  a  suitable  notation.  When  such  a  nota- 
tion was  adopted  at  the  close  of  the  Middle  Ages,  the 
art  advanced  rapidly.  That  advance  was  one  feature 
of  the  Renaissance,  a  detail  in  the  great  intellectual 
awakening  of  that  marvelous  half  century  from  1450 
to  1500,  "the  age  of  progress." 

The  choice  between  the  old  and  the  new  in  arith- 

66 


ARITHMETIC   IN    THE  RENAISSANCE.  67 

metical  notation  is  well  pictured  by  the  illustration*  of 
arithmetic  in  the  first  printed  cyclopedia,  the  Marga- 
rita Philosophica  (1503).  Two  accountants  are  at 
their  tables.  The  old  man  is  using  the  abacus ;  the 
voting  man,  the  Hindu  numerals  so  familiar  to  us. 
The  aered  reckoner  looks  askance  at  his  youthful  rival, 
in  whose  face  is  hope  and  confidence  ;  while  on  a  dais 
behind  both  stands  the  goddess  to  decide  which  shall 
have  the  ascendency.  Her  eyes  are  fixed  on  the 
younger  candidate,  at  her  right,  and  there  can  be  no 
doubt  that  to  the  new  numerals  is  to  be  the  victory. 
The  background  of  the  picture  is  characteristically 
medieval.  It  is  an  apt  illustration  of  the  passing  of 
the  old  arithmetic.  To  us  of  four  centuries  after,  it 
whispers  (as  one  has  said  of  the  towers  of  old  Ox- 
ford) "the  last  enchantment  of  the  Middle  Age." 

The  anonymous  book  known  as  the  Treviso  arith- 
metic, from  its  place  of  publication,  is  the  first  arith- 
metic ever  printed.  It  appeared  in  1478.  In  this 
Italian  work  of  long  ago  the  multiplication  looks  mod- 
ern. But  long  division  was  by  the  galley  (or  "scratch") 
method  then  prevalent. 

Paciolrs  Summa  di  Arithmetical  was  published  in 
1494  (some  say  ten  years  earlier).  It  also  uses  the 
Hindu  numerals. f 

Tonstall's  arithmetic  (1522)  was  "the  first  impor- 
tant arithmetical  work  of  English  authorship. "J  De 
Morgan  calls  the  book  "decidedly  the  most  classical 
which  ever  was  written  on  the  subject  in  Latin,  both 
in  purity  of  style  and  goodness  of  matter." 

*  See  frontispiece. 

fin  Pacioli's  work,  the  words  "zero"  (cero)  and  "million" 
(millione)  are  found  for  the  first  time  in  print.  Cantor,  II, 
284. 

t  Cajori,  'Hist,  of  Elem.  Math.,  p.  180. 


68       A  SCRAP-BOOK  OF  ELEMENTARY   MATHEMATICS. 

Recorde's  celebrated  Grounde  of  Aries  (1540)  was 
written  in  English.  It  uses  the  Hindu  numerals,  but 
teaches  reckoning  by  counters.  The  exposition  is  in 
dialogue  form. 

The  first  English  work  on  double  entry  book-keep- 
ing, by  John  Mellis  (London,  1588),  has  an  appendix 
on  arithmetic. 

The  Pathzvay  to  Knowledge,  anonymous,  translated 
from  Dutch  into  English  by  W.  P.,  was  published  in 
London  in  1596.  It  contains  two  lines  which  are  im- 
mortal. The  translator  has  been  said  to  be  the  author 
of  the  lines.  In  modernized  form  they  are  known  to 
every  schoolboy.  Of  all  the  arithmetical  doggerel  of 
that  age,  this  is  pre-eminently  the  classic : 

"Thirtie  daies  hath  September,  Aprill,  June,  and  No- 
vember, 
Februarie,  eight  and  twentie  alone  ;  all  the  rest  thirtie 
and  one." 

On  the  subject  of  early  arithmetics  De  Morgan's 
Arithmetical  Books  is  the  standard  work.  An  inter- 
esting contribution  to  the  subject  is  Prof.  David  Eu- 
gene Smith's  illustrated  article,  "The  Old  and  the  New 
Arithmetic,"  published  by  Ginn  &  Co.  in  their  Text- 
book Bulletin,  February.  1905.- 


NAPIER'S  RODS  AND  OTHER  MECHANICAL 
AIDS  TO  CALCULATION. 

No  mathematical  invention  to  facilitate  computation 
has  been  made  for  three  centuries  that  is  comparable 
to  logarithms.  Napier's  rods,  or  "Napier's  bones," 
once  famous,  owe  their  interest  now  largely  to  the 
fact  that  they  are  the  invention  of  the  man  who  gave 
logarithms  to  the  world,  John  Napier,  baron  of  Mer- 
chiston.  The  inventor's  description  of  the  rods  is  con- 
tained in  his  Rabdologia,  published  in  1617,  the  year 
of  his  death. 

The  rods  consist  of  10  strips  of  wood  or  other  ma- 
terial, with  square  ends.  A  rod  has  on  each  of  its 
four  lateral  faces  the  multiples  of  one  of  the  digits. 
One  of  the  rods  has,  on  the  four  faces  respectively, 
the  multiples  of  0,  1,  9,  8;  another,  of  0,  2,  9,  7;  etc. 
Each  square  gives  the  product  of  two  digits,  the  two 
figures  of  the  product  being  separated  by  the  diag- 
onal of  the  square.  E.  g.,  in  Fig.  2  the  lowest  right 
hand  square  contains  the  digits  7  and  2,  72  being  the 
product  of  9  (at  the  left  of  the  same  row)  and  8  (at 
the  top  of  the  rod). 

Fig.  2  represents  the  faces  of  the  rods  giving  the 
multiples  of  4,  3  and  8,  placed  together  and  against  a 
rod  containing  the  nine  digits  to  be  used  as  multiplier, 
all  in  position  to  multiply  438  by  any  number — say 


69 


yO      A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

26.  The  products  are  written  off,  from  the  rods.  But 
the  tens  digit  in  each  case  is  to  be  added  to  the  next 
units  digit;  that  is,  the  two  figures  in  a  rhomboid  are 
to  be  added.    The  operation  of  multiplying  438  by  26, 


Fig.  i.* 


after  arranging  the  rods  as  in  Figure  2,  would  be 
somewhat  as  follows :  beginning  at  the  right  hand  and 
multiplying  first  by  6,  we  have  8,  4  +  8,  (carrying  the 

*  From  Lucas,  III,  76. 


NAPIER  S   RODS. 


71 


1  )  1  4.  1  +  4^  2,  giving  the  number  (from  left  to  right) 
2628,  the  first  partial  product.     Similarly  876  is  read 

from  the  row  of  squares  at  the 
right  of  the  multiplier  2.  It  is 
shifted  one  place  to  the  left  in 
writing  it  under  the  former  par- 
tial product.  Then  these  two 
numbers  are  added. 

Somewhat   analogous   is  the 
use  of  the  rods  for  division. 

"It  is  evident  that  they  would 
be  of  little  use  to  any  one  who 
knew  the  multiplication  table  as 
far  as  9  x  9."*  Though  pub- 
lished (and  invented)  later  than 
logarithms,  which  we  so  much 
admire,  the  rods  were  welcomed 
more  cordially  by  contempo- 
raries. Several  editions  of  the 
Rabdologia  were  brought  out 
on  the  Continent  within  a  de- 
cade. "Nothing  shows  more  clearly  the  rude  state  of 
arithmetical  knowledge  at  the  beginning  of  the  seven- 
teenth century  than  the  universal  satisfaction  with 
which  Napier's  invention  was  welcomed  by  all  classes 
and  regarded  as  a  real  aid  to  calculation."*  It  is  from 
this  point  of  view  that  the  study  of  the  rods  is  inter- 
esting and  instructive  to  us. 

The  Rabdologia  contains  other  matter  besides  the 
description  of  rods  for  multiplication  and  division. 
But  such  mechanical  aids  to  calculation  are  soon  super- 
seded. 


I 

4 

3 

8 

2 

/Z 

/(y 

1  / 

/6 

3 

1  / 
/7 

/  9 

2/ 
/  4 

4 

I  / 

/  6 

/    2 

/  2 

5 

2 / 

/  Q 

/  5 

/  0 

6 

2/ 

/  4 

1  / 
/  8 

4/ 
/  8 

7 

2  / 
/  8 

2  / 

"A 

8 

3/ 
/  2 

2/ 
/  4 

6/ 
/  4 

9 

3/ 
/    6 

2  / 
/  7 

v7 

/  2 

Fig.  2. 


*  Dr.   Glaisher  in  his  article  "Napier"   in  the  Encyclopedia 
Britannic  a. 


*J2       A  SCRAP-BOOK  OF  ELEMENTARY   MATHEMATICS. 

It  is  worthy  of  note  in  this  connection,  however,  that 
in  the  absence  of  so  facile  an  instrument  for  calcula- 
tion as  our  Arabic  notation,  simple  mechanical  devices 
might  be  found  so  serviceable  as  to  persist  for  cen 
turies.  The  abacus,  which  is  familiar  to  almost  every 
one,  but  only  as  a  historical  relic,  a  piece  of  illustra- 
tive apparatus,  or  a  toy,  was  a  highly  important  aid  to 
computation  among  the  Greeks  and  Romans.  Similar 
to  the  abacus  is  the  Chinese  swan  pan.  It  is  said  that 
Oriental  accountants  are  able,  by  its  use,  to  make  com- 
putations rivaling  in  accuracy  and  speed  those  per- 
formed by  Occidentals  with  numerals  on  paper. 

Modern  adding  machines,  per  cent  devices,  and  the 
more  complicated  and  costly  calculating  instruments 
have  led  up  to  such  mechanical  marvels  as  "electrical 
calculating  machines"  and  the  machines  of  Babbage 
and  Scheutz,  which  latter  prepare  tables  of  logarithms 
and  of  logarithmic  functions  without  error  arithmetical 
or  typographical,  computing,  stereotyping  and  deliv- 
ering them  ready  for  the  press. 

If  Napier's  rods  be  regarded  as  exemplars  of  such 
products  of  the  nineteenth  century,  they  are  primitive 
members  of  a  long  line  of  honorable  succession. 


AXIOMS   IN   ELEMENTARY  ALGEBRA. 

Many  text-books  on  the  subject  introduce  equations 
with  a  list  of  axioms  such  as  the  following: 

1.  Things  equal  to  the  same  thing  or  equal  things 
are  equal  to  each  other. 

2.  If  equals  be  added  to  equals,  the  sums  are  equal. 

3.  If  equals  be  subtracted  from  equals,  the  remain- 
ders are  equal. 

4.  If  equals  be  multiplied  by  equals,  the  products 
are  equal. 

5.  If  equals  be  divided  by  equals,  the  quotients  are 
equal. 

6.  The  whole  is  greater  than  any  of  its  parts. 

7.  Like  powers,  or  like  roots,  of  equals  are  equal. 

These  time-honored  "common  notions"  are  the  foun- 
dation of  logical  arithmetic.  On  them  is  based  also 
the  reasoning  of  algebra.  But  it  is  most  desirable 
that,  when  we  extend  their  meaning  to  the  comparison 
of  algebraic  numbers,  we  should  notice  the  limitations 
of  the  axioms.  Generalization  is  a  characteristic  of 
mathematics.  When  we  generalize,  we  remove  limi- 
tations that  have  been  stated  or  implied.  A  proposi- 
tion true  with  those  limitations  may  or  may  not  be 
true  without  them.  For  illustration:  When  we  pro- 
ceed from  geometry  of  two  dimensions  to  geometry 
of  three  dimensions,  the  limitation,  always  understood 
in  plane  geometry,  that  all  figures  considered  are  (ex- 
cept while  employing  the  motion  postulate  for  super- 

73 


74       A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

position)  in  the  plane  of  the  paper  or  blackboard,  is 
removed.  Some  of  the  propositions  true  in  plane 
geometry  hold  also  in  solid,  and  some  do  not.  Com- 
pare in  this  respect  the  two  theorems,  "Through  a 
given  external  point  only  one  perpendicular  can  be 
drawn  to  a  given  line,''  and,  "Through  a  given  internal 
point  only  one  perpendicular  can  be  drawn  to  a  given 
line."*  For  another  illustration  see  the  paragraph  (p. 
37),  "Are  there  more  than  one  set  of  prime  factors 
of  a  number?"  No  when  factor  means  arithmetic 
number ;  yes  when  the  meaning  of  the  word  is  extended 
to  include  complex  numbers.  See  also  instances  of 
the  "fallacy  of  accident,"  p.  85  f. 

We  might  expect  that  some  of  the  axioms  of  arith- 
metic would  need  qualification  when  we  attempt  to  ex- 
tend them  so  as  to  apply  to  algebraic  numbers.  And 
that  is  what  we  find.  But  we  do  not  find  that  all 
authors  have  notified  their  readers  of  the  limitations 
or  have  observed  them  in  their  own  use  of  the  axioms. 
Surely  it  is  not  too  much  to  expect  that  the  axioms  of 
a  science  shall  be  true  and  applicable  in  the  sense  in 
which  the  terms  are  used  in  that  science. 

The  fifth,  or  "division  axiom,"  should  receive  the 
important  qualification  given  it  by  the  best  of  the 
books,  "divided  by  equals,  except  zero.3'  Without  such 
limitation  the  statement  is  far  from  axiomatic. 

A  writer  of  the  sixth  "axiom"  may  also  have  +  7 
on  another  page  something  like  this :  "+  3  is  the  -  5 
whole,  or  sunt/'  Seeing  that  one  of  its  parts  is  +2 
+  7,  one  wonders  how  the  author,  in  a  text-book  -  1 
on  algebra,  could  ever  have  written  the  "ax-  +  3 
iom,"  "The  whole  is  greater  than  any  of  its  parts." 

*  Using  the  term  perpendicular  in  the  sense  customary  in 
elementary  geometry. 


AXIOMS  IN  ELEMENTARY  ALGEBRA.  75 

In  the  seventh  axiom,  like  roots  of  equals  are  equal 
arithmetically.  Otherwise  worded :  Like  real  roots 
of  equals  are  equal,  like  signs  being  taken.'" 

When  we  use  the  word  "equal"  in  the  axioms,  do 
we  mean  anything  else  than  "same" — If  two  numbers 
are  the  same  as  a  third  number,  they  are  the  same  as 
each  other,  etc.? 

*  The  defense  often  heard  for  the  unqualified  axiom,  Like 
roots  of  equals  are  equal,  in  algebra — that  like  here  means 
equal — would  reduce  the  axiom  to  a  platitude,  Roots  are 
equal  if  they  are  equal.  Besides  being  insipid,  this  is  in- 
sufficient. To  be  of  any  use,  the  axiom  must  mean,  that  if  C 
and  D  are  known  to  represent  each  a  square  root,  or  each  a 
cube  root,  of  A  and  B  respectively,  and  if  A  and  B  are  known 
to  be  equal,  then  C  and  D  are  as  certainly  known  to  be  two 
expressions  for  the  same  number.  Now  in  the  case  of  square 
roots  this  inference  is  justified  only  when  like  signs  are  taken. 

For  cube  roots,  if  A=B=1,  then — t^  +  t^v/ — 3  is  a  cube  root 
of  A,  and—  —  —  ^s/—  3  is  a  cube  root  of  B;  but  —  ?  +  -\/— 3 

and  — -= — Tj-  >/ — 3  are  not  expressions  for  the  same  number. 

If  their  modulus  (page  94)  be  taken  as  their  absolute  value, 
they  are  equal  to  each  other  and  to  the  real  cube  root  1  in 
absolute  value.  If  our  axiom  be  made  to  read,  Like  odd  real 
roots  are  equal,  it  is  applicable  to  such  roots  without  trouble. 
A  has  but  one  cube  root  that  is  real,  and  B  has  but  one,  and 
they  are  equal. 

It  is  interesting  to  notice  in  passing  that  the  two  numbers 

just   used, — «+-«%/— 3  and   —  —  —  -?-\/-— 3,   are   a   pair   of 

unequal  numbers  each  of  which  is  the  square  of  the  other. 


DO  THE  AXIOMS  APPLY  TO   EQUATIONS? 

Most  text-books  in  elementary  algebra  use  them  as 
if  they  applied.  Most  of  the  algebras  have,  somewhere 
in  the  first  fifty  or  sixty  pages,  something  like  this: 

3*  +  4=19 

Subtracting  4  from  each  member, 

3x  =  15  Ax.  3 

Dividing  by  3, 

x  =  5  Ax.  5 

This  shows  how  common  some  very  loose  thinking 
on  this  subject  is.  For  although  no  mistake  has  been 
made  in  the  algebraic  operation,  the  citation  of  axioms 
as  authority  for  these  steps  opens  the  way  for  a  pupil 
to  divide  both  members  of  an  equation  by  an  unknown, 
in  which  case  he  drops  a  solution, *  or  to  apply  one  of 
the  other  axioms  and  introduce  a  solution. 

As  a  matter  of  fact,  the  axioms  do  not  apply  directly 
to  equations:  for  (A)  one  can  follow  the  axioms, 
make  no  mistake,  and  arrive  at  a  result  which  is  in- 
correct ;  (B)  he  can  violate  the  axioms  and  come  out 
right;  (C)  the  axioms,  from  their  very  nature,  can 
not  apply  directly  to  equations. 

*  Every  teacher  of  elementary  algebra  is  aware  of  the 
tendency  of  pupils  (unless  carefully  guided)  to  "divide  through 
by  x"  when  possible,  and  to  fail  to  note  that  they  have  lost  out 
the  solution  x  =  o. 


76 


DO   THE   AXIOMS   APPLY   TO   EQUATIONS?  JJ 

(A)  To  follow  axioms  and  come  out  wrong'. 

.,--1=2  (1) 

Multiplying  each  member  by  x  —  $f 

.r2-&r  +  5  =  2.r-10  Ax.  4 

Subtracting  x  -  7  from  each  member, 

x2-7x+\2  =  x-3  Ax.  3 

Dividing  each  member  by  x  —  3, 

4T-4=1  Ax.  5 

Adding  4  to  each  member, 

.r  =  5  Ax.  2 

But  x  -  5  does  not  satisfy  ( 1 ) .     The  only  value  of  x 
that  satisfies    (1)    is  3. 

Misunderstanding  at  this  point  is  so  common  that 
it  is  deemed  best  to  be  explicit  at  the  risk  of  being 
tedious.  The  multiplication  by  x  -  5  introduces  the 
solution  x  =  5,  and  the  division  by  x  -  3  loses  the  solu- 
tion x  -  3.  Now  it  may  be  argued,  that  the  axioms 
of  the  preceding  section  when  properly  qualified  ex- 
clude division  by  zero,  and  that  x  -  3  is  here  a  form 
of  zero  since  3  is  the  value  of  x  for  which  equation 
( 1 )  is  true.  Exactly ;  but  this  only  shows  that  in 
operating  with  equations  the  question  for  what  value 
of  x  they  are  true  is  bound  to  be  raised.  The  attempt 
to  qualify  the  axioms  and  adjust  them  to  this  necessity 
will,  if  thoroughgoing,  lead  to  principles  of  equivalency 
of   equations.*      Any   objector   is   requested   to   study 

*  Such,  for  example,  as  the  following : 

To  add  or  subtract  the  same  expression  (known  or  unknown) 
to  both  members  of  an  equation,  does  not  affect  the  value  of  x 
(the  resulting  equation  is  equivalent  to  the  original). 

To  multiply  or  divide  both  members  by  a  known  number  not 
zero,  does  not  affect  the  value  of  x. 

To  multiply  or  divide  both  members  by  an  integral  function 
of  x,  introduces  or  loses,  respectively,  solutions  (namely,  the 
solution  of  the  equation  formed  by  putting  the  multiplier 
equal  to  zero)  it  being  understood  that  the  equations  are  in 
the  standard  form. 


?8       A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

carefully  the  principles  of  equivalency  as  set  forth  in 
one  of  the  best  algebras  and  notice  their  relation  to  the 
axioms  on  the  one  hand  and  to  operations  with  equa- 
tions on  the  other,  and  see  whether  he  is  not  then  pre- 
pared to  say  that  the  axioms  do  not  apply  directly  to 
equations. 

It  should  be  noted  that  the  foregoing  is  not  an 
attack  on  the  integrity  of  the  axioms,  but  only  on  the 
application  of  them  where  they  are  not  applicable. 

If  it  be  objected  that  in  (A)  the  axioms  are  not  really 
followed,  the  reply  is,  that  they  are  here  followed  as 
they  are  naturally  followed  by  pupils  taught  to  apply 
them  directly  to  equations,  and  as  they  are  occasionally 
followed  by  the  authors  of  some  elementary  algebras, 
only  the  errors  are  here  made  more  glaring  and  the 
process  reduced  ad  absurdum. 

(B)  To  violate  the  axioms  and  come  out  right: 

In  order  to  avoid  the  objection  that  the  errors  made 
by  violating  two  axioms  may  just  balance  each  other, 
only  one  axiom  will  be  violated. 

x-\=2  (1) 

Add  10  to  one  member  and  not  to  the  other.  This  will 
doubtless  be  deemed  a  sufficiently  flagrant  transgres- 
sion of  the  "addition  axiom": 

^•  +  9  =  2  (2) 

Multiplying  each  member  by  x  -  3, 

x2  +  6x -27  =  2x-6       (3)  Ax.  4 

Subtracting  2x-6  from  each  member, 

.r*  +  4.r-21=0  (4)  Ax.  3 

Dividing  each  member  by  x  +  7, 

.r-3  =  0  (5)  Ax.  5 

Adding  3  to  each  member, 

x  =  3  Ax.  2 


DO  THE  AXIOMS   APPLY  TO   EQUATIONS?  79 

Inasmuch  as  3  is  the  correct  root  of  equation  (1),  the 
error  in  the  first  step  must  have  been  balanced  by  an- 
other, or  by  several.  It  was  done  in  obtaining  (3)  and 
(5),  though  at  both  steps  the  axioms  were  applied. 

(C)  The  axioms,  from  their  very  nature,  can  not 
have  any  direct  application  to  equations. 

The  axioms  say  that — if  equals  be  added  to  equals 
etc. — the  results  are  equal.  But  the  question  in  solv- 
ing equations  is,  For  what  value  of  x  are  they  equal? 
Of  course  they  are  equal  for  some  value  of  x.  So 
when  something  was  added  to  one  member  and  not 
to  the  other,  the  results  were  equal  for  some  value  of 
x.  Arithmetic,  dealing  with  numbers,  needs  to  know 
that  certain  resulting  numbers  are  equal  to  certain 
others ;  but  algebra,  dealing  with  the  equation,  the 
conditional  equality  of  expressions,  needs  to  know  on 
zvhat  condition  the  expressions  represent  the  same 
number — in  other  words,  for  what  values  of  the  un- 
known the  equation  is  true.  In  (B)  above,  the  ob- 
jection to  equation  (2)  is  not  that  its  two  members 
are  not  equal  (they  are  "equal"  as  much  as  are  the 
two  members  of  the  first  equation)  but  that  they  are 
not  equal  for  the  same  value  of  x  as  in  the  first  equa- 
tion;  that  is  (2)  is  not  equivalent  to  (1). 

The  principles  of  equivalency  of  equations  as  given 
in  a  few  of  the  best  of  the  texts  are  not  too  difficult 
for  the  beginner.  The  proof  of  them  may  well  be  de- 
ferred till  later.  Even  if  never  proved,  they  would 
be,  for  the  present  purpose,  vastly  superior  to  axioms 
that  do  not  apply.  To  give  no  reasons  would  be  pref- 
erable to  the  practice  of  quoting  axioms  that  do  not 
apply. 


So       A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

The  axioms  have  their  place  in  connection  with 
equations ;  namely,  in  the  proof  of  the  principles  of 
equivalency.  To  apply  the  axioms  directly  in  the  so- 
lution of  equations  is  an  error. 

Pupils  can  hardly  be  expected  to  think  clearly  about 
the  nature  of  the  equation  when  they  are  so  misled. 
How  the  authors  of  the  great  majority  of  the  elemen- 
tary texts  can  have  made  so  palpable  a  mistake  in  so 
elementary  a  matter,  is  one  of  the  seven  wonders  of 
algebra. 


CHECKING  THE   SOLUTION   OF  AN   EQUA- 
TION. 

The  habit  which  many  high-school  pupils  have  of 
checking  their  solution  of  an  equation  by  first  sub- 
stituting for  x  in  both  members  of  the  given  equa- 
tion, performing  like  operations  upon  both  members 
until  a  numerical  identity  is  obtained,  and  then  de- 
claring their  work  "proved/'  may  be  illustrated  by  the 
following  "proof,"  in  which  the  absurdity  is  appa- 
rent :  Syi 


1+  Vr+2  =  1-  <\2-x 

(1) 

Solution 

^x+2=  -  V12-* 

(2) 

*+2=*12-.r 

(3) 

2*  =  10 

x  =  5 

"Proof" 

1+ V5  +  2  =  l-  Vl2-5 
V5  +  2=-  V12-5 

- 

5  +  2  =  12-5 

7  =  7 

Checking  in  the  legitimate  manner — by  substituting 
in  one  member  of  the  given  equation  and  reducing  the 
resulting  number  to  its  simplest  form,  then  substituting 
in  the  other  member  and  reducing  to  simplest  form — 
we  have  1  +  y7  for  the  first  member,  and  1  -  V  7  for 
the  second.  As  these  are  not  equal  numbers,  5  is  not 
a  root  of  the  equation.     There  is  no  root. 

81 


82      A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

'  The  5  was  introduced  in  squaring.  That  is,  #  =  5 
satisfies  equation  (3)  but  not  (2)  or  (1).  By  the 
change  of  a  sign  in  either  (1)  or  (2)  we  obtain  an 
equation  that  is  true  for  x  =  5: 

1+  V*+2 :  =  1+  V12-* 

When  rational  equations  are  derived  from  irrational 
by  involution,  there  are  always  other  irrational  equa- 
tions, differing  from  these  in  the  sign  of  a  term,  from 
which  the  same  rational  equations  would  be  derived. 
In  a  popular  algebra  may  be  found  the  equation 

x+5-  Vjtr+5  =  6 
and  in  the  answer  list  printed  in  the  book,  "4,  or  -  1" 
is  given  for  this  equation.     4  is  a  solution,  but  -  1  is 
not.     Unfortunately  this  instance  is  not  unique. 

As  the  fallacy  in  the  erroneous  method  shown  above 
is  in  assuming  that  all  operations  are  reversible,  that 
method  may  be  caricatured  by  the  old  absurdity, 

To  prove ' that  5  =  1  \J 
Subtracting  3  from  each,  2  =  -2/^ 
Squaring  4  =  4 

,-.5  =  1! 


ALGEBRAIC  FALLACIES. 

A  humorist  maintained  that  in  all  literature  there 
are  really  only  a  few  jokes  with  many  variations,  and 
proceeded  to  give  a  classification  into  which  all  jests 
could  be  placed — a  limited  list  of  type  jokes.  A  fellow 
humorist  proceeded  to  reduce  this  number  (to  three, 
if  the  writer's  memory  is  correct).  Whereupon  a 
third  representative  of  the  profession  took  the  remain- 
ing step  and  declared  that  there  are  none.  Whether 
these  gentlemen  succeeded  in  eliminating  jokes  alto- 
gether or  in  adding  another  to  an  already  enormous 
number,  depends  perhaps  on  the  point  of  view. 

The  writer  purposes  to  classify  and  illustrate  some 
of  the  commoner  algebraic  fallacies,  in  the  hope,  not 
of  adding  a  striking  original  specimen,  but  rather  of 
standardizing  certain  types,  at  the  risk  of  blighting 
them.  Fallacies,  like  ghosts,  are  not  fond  of  light. 
Analysis  is  perilous  to  all  species  of  the  genus. 

Of  the  classes,  or  subclasses,  into  which  Aristotle 
divided  the  fallacies  of  logic,  only  a  few  merit  special 
notice  here.  Prominent  among  these  is  that  variety 
of  paralogism  known  as  undistributed  middle.  In 
mathematics  it  masks  as  the  fallacy  of  converse,  or 
employing  a  process  that  is  not  uniquely  reversible 
as  if  it  were.     For  example,  the  following:* 

*  Taken,  with  several  of  the  other  illustrations,  from  the 
fallacies  compiled  by  W.  W.  R.  Ball.  See  his  Mathematical 
Recreations  and  Essays. 

83 


84       A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

Let  c  be  the  arithmetic  mean  between  two  unequal 
numbers  a  and  b  ;  that  is,  let 

a  +  b  =  2c  (1) 

Then  (a  +  b)  (a-d)=2c(a-d) 

a2  —  b2  =  2ac  —  2bc 
Transposing,  a2  —  2ac  =  b2  —  2bc 

Adding  c2  to  each,       a2-2ac+c2  =  b2-2bc+c2      (2) 

.'  .a—c  =  b  —  c  (3) 

and        a  =  b 
But  a  and  b  were  taken  unequal. 

Of  course  the  two  members  of  (3)  are  arithmetic- 
ally equal  but  of  opposite  quality;  their  squares,  the 
two  members  of  (2),  are  equal.  The  fallacy  here  is 
so  apparent  that  it  would  seem  superfluous  to  expose 
it,  were  it  not  so  common  in  one  form  or  another. 

For  another  example  take  the  absurdity  used  in  the 
preceding  section  to  caricature  an  erroneous  method  of 
checking  a  solution  of  an  equation.  Let  us  resort  to 
a  parallel   column  arrangement: 

A  bird  is  an  animal;       -  Two     equal     numbers     have 

equal  squares; 

A  horse  is  an  animal;  These     two     numbers     have 

equal  squares; 
.•.  A  horse  is  a  bird.  .•.  These  two  numbers  are  equal. 

The  untutored  man  pooh-  The   first-year   high-school 

poohs    at    this,    because    the'  pupil   derides   this   whenever 

conclusion  is    absurd,  but  fails  the  conclusion  is  absurd,  but 

to  notice  a  like  fallacy  on  the  would  allow  to  pass  unchal- 

lips  of  the  political   speaker  lenged  the  fallacious  method 

of  his  own  party.  of  checking  shown  in  the  pre- 
ceding section. 

In  case  of  indicated  square  roots  the  fallacy  may  be 
much  less  apparent.  By  the  common  convention  as 
to  sign,  +  is  understood  before  V-     Considering,  then. 


ALGEBRAIC  FALLACIES.  85 

only  the  positive  even  root  or  the  real  odd  root,  it  is 
true  that  "like  roots  of  equals  are  equal,"  and 

v  ab=  v  a-  v  b 

But  if  a  and  b  are  negative,  and  n  even,  the  identity 
no  longer  holds,  and  by  assuming  it  we  have  the  ab- 
surdity 

V(_l)  (-1)=  V^T.  V-l 
1=-1 


4 


V, 


a 


Or  take  for  granted  that  x\-  =  -7- for  all  values  of 

\b       Vb 

the  letters.     The  following  is  an  identity,  since  each 

member  =  V— 1: 


Hence ! 


VI      V-i 


Clearing  of  fractions,      (  Vl )  2=  (  V— 1 ) 2 
Or  1=-1 

The  "fallacy  of  accident,"  by  which  one  argues 
from  a  general  rule  to  a  special  case  where  some  cir- 
cumstance renders  the  rule  inapplicable,  and  its  con- 
verse fallacy,  and  De  Morgan's  suggested  third  variety 
of  the  fallacy,  from  one  special  case  to  another,  all 
find  exemplification  in  pseudo-algebra.  As  a  general 
rule,  if  equals  be  divided  by  equals,  the  quotients  are 
equal ;  but  not  if  the  equal  divisors  are  any  form  of 
zero.  The  application  of  the  general  rule  to  this  special 
case  is  the  method  underlying  the  largest  number  of 
the  common  algebraic  fallacies. 


86      A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 


•">  9  O 

x-  -  x-  -  x-  -  X 


Factoring  the  first  member  as  the  difference  of  squares, 
and  the  second  by  taking  out  a  common  factor, 

(x  +  x)  (x-x)  =x(x-x)  (1) 

Canceling  x-x,  x  +  x=x  (2) 

2x  =  x 
2  =  1  (3) 

Dividing  by  0  changes  identity  (1)  into  equation  (2), 
which  is  true  for  only  one  value  of  x,  namely  0.     Di- 
viding (2)  by  x  leaves  the  absurdity  (3). 
Take  another  old  illustration:* 


Let 

X- 

=  1 

/ 

Then 

X2  = 

-X 

And 

X2- 

-1  = 

-  x- 

1 

Dividing 

botr 

i  by  x  -  1 , 

x+l- 

-l^ 

But 

X- 

=  1 

Whence, 

bv 

substituting, 

2  = 

=  1 

The  use  of  a  divergent  series  furnishes  another 
type  of  fallacy,  in  which  one  assumes  something  to 
be  true  of  all  series  which  in  fact  is  true  only  of  the 
convergent.  For  this  purpose  the  harmonic  series  is 
perhaps  oftenest  employed. 

i+|+|+|  +  ... 

Group  the  terms  thus: 

1       /l    .   1\       /l    .  1    .1    .  1 


-  +  .  .  .to  8  terms)  +  (~-\ 

Every  term   (after  the  second)   in  the  series  as  now 
written  >}/2.     Therefore  the  sum  of  the  first  n  terms 

*  Referred  to  by  De  Morgan  as  "old"  in  a  number  of  the 
Athcnccum  of  forty  years  ago. 


ALGEBRAIC  FALLACIES.  87 

increases   without   limit  as   n    increases   indefinitely.* 

The  series  has  no  finite  sum ;  it  is  divergent.     But  if 

the  signs  in  this  series  are  alternately  +  and  -    the 
series 

2^3      4+5 
is  convergent.  With  this  in  mind,  the  following  fallacy 
is  transparent  enough : 

10*2-1-!+!-!+!-!+... 


"'1  +  3 


H+.. .HK+I+...) 
.[(:+!+!+...)+(!+!+!+...)] 


-<!+!+!+. 


••) 


-0 
But  log  1  =  0 
Suppose  »  written  in  place  of  each  parenthesis. 

30  and  0  are  both  convenient  "quantities"  for  the 
fallacy  maker. 

By  tacitly  assuming  that  all  real  numbers  have  loga- 
rithms and  that  they  are  amenable  to  the  same  laws 
as  the  logarithms  of  arithmetic  numbers,  another  type 
of  fallacv  emerges: 

(-d2=i 

Since  the  logarithms  of  equals  are  equal, 
2  log  (-l)=logl  =  0 
/.log  (-1)=0 
/.log  (-l)=logl 
and    -1   =1 
*The  sum  of  the  first  2"  terms  >l  +  !4«. 


88      A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

The  idea  of  this  type  is  credited  to  John  Bernoulli. 
Some  great  minds  have  turned  out  conceits  like  these 
as  by-products,  and  many  amateurs  have  found  de- 
light in  the  same  occupation.  To  those  who  enjoy 
weaving  a  mathematical  tangle  for  their  friends  to 
unravel,  the  diversion  may  be  recommended  as  harm- 
less. And  the  following  may  be  suggested  as  prom- 
ising points  around  which  to  weave  a  snarl:  the  tan- 
gent of  an  angle  becoming  a  discontinuous  function 
for  those  particular  values  of  the  angle  which  are  rep- 
resented by  (n  +  ^)<7r;  discontinuous  algebraic  func- 
tions; the  fact  that  when  h,  j  and  k  are  rectangular 
unit  vectors  the  commutative  law  does  not  hold,  but 
hjk  -  -  kjh  ;  the  well-known  theorems  of  plane  geom- 
etry that  are  not  true  in  solid  geometry  without  quali- 
fication ;  etc. 

Let  us  use  one  of  these  to  make  a  fallacy  to  order. 
In  the  fraction  1/x,  if  the  denominator  be  diminished, 
the  fraction  is  increased. 

When  x  —  S,  3,  1,  -  1,  -3,  -5,  a  decreasing  series, 

then  l/x  =  1/5,  1/3,  1,  -  1,  -  1/3,  -  1/5,  an  increas- 
ing series, 
as,  by  rule,  each  term  of  the  second  series  is  greater 
than  the  term  before  it:  1/3 >  1/5,  l>l/3,  -l/5>— 1/3. 
Then  the  fourth  term  is  greater  than  the  third ;  that  is 

-1>  +  1. 

Neither  the  fallacies  of  formal  logic  nor  those  of 
algebra  invalidate  sound  reasoning.  From  the  coun- 
terfeit coin  one  does  not  infer  that  the  genuine  is  value- 
less. Scrutiny  of  the  counterfeit  may  enable  us  to 
avoid  being  deceived  later  by  some  particularly  clever 
specimen.  Counterfeit  coins  also,  if  so  stamped,  make 
good  playthings. 


TWO  HIGHEST  COMMON  FACTORS. 

If  asked  for  the  H.C.F.  of  a2  -  x2  and  x3-a3,  one 
pupil  will  give  a  -  x,  and  another  x  -  a.  Which  is 
right?  Both.  It  is  only  in  such  a  case  that  pupils 
raise  the  question ;  but  the  example  is  not  peculiar  in 
having  two  H.C.F.  If  the  given  expressions  had  been 
x2  -  a2  and  x3  -a3,  x-a  would  naturally  be  obtained, 
and  would  probably  be  the  only  H.C.F.  offered ;  but 
a  -  x  is  as  much  a  common  factor  and  is  of  as  high 
a  degree.  Perhaps  it  is  taken  as  a  matter  of  course — 
certainly  it  is  but  rarely  stated — that  every  set  of  al- 
gebraic expressions  has  two  highest  common  factors, 
arithmetically  equal  but  of  opposite  quality. 

As  the  term  "highest"  is  used  in  a  technical  way, 
the  purist  will  perhaps  pardon  the  solecism  "two 
highest." 

Similarly,  of  course,. there  are  two  L.C.M.  of  every 
set  of  algebraic  expressions.  By  going  through  the 
answer  list  for  exercises  in  L.C.M.  in  an  algebra  and 
changing  the  signs,  one  obtains  another  list  of  an- 
swers. 


*  POSITIVE  AND  NEGATIVE  NUMBERS. 

To  speak  of  arithmetical  numbers  as  positive,  is 
still  so'  common  an  error  as  to  need  correction  at  every 
opportunity.  The  numbers  of  arithmetic  are  not  posi- 
tive. They  are  numbers  without  quality.  Negatives 
are  not  later  than  positives,  either  in  the  individual's 
conception  or  in  that  of  the  race.  How  can  the  idea 
of  one  of  two  opposites  be  earlier  than  the  other,  or 
clearer?  The  terms  "positive"  and  "negative"  being 
correlative,  neither  can  have  meaning  without  the 
other.* 

An  "algebraic  balance"  has  been  patented  and  put 
on  the  market,f  designed  to  illustrate  positive  and 
negative  numbers,  also  transposition  and  the  other 
operations  on  an  equation.  It  is  composed  of  a  sys- 
tem of  levers  and  scale  pans  with  weights.  The  value 
of  this  excellent  apparatus  in  ijlustrating  positive  and 
negative  numbers  is  in  showing  them  to  be  opposites 
of  each  other.  E.  g.,  a  weight  in  the  positive  scale 
pan  neutralizes  the  pull  on  the  beam  exerted  by  a 

*  A  good  exercise  to  develop  clear  thinking  as  to  the  rela- 
tion between  positive,  negative  and  arithmetic  numbers  is,  to 
consider  the  correspondence  of  the  positive  and  negative  solu- 
tions of  an  equation  to  the  arithmetic  solutions  of  the  problem 
that  gave  rise  to  the  equation,  and  the  question  to  what 
primary  assumptions  this  correspondence  is  due. 

t  By  P,  C.  Donecker,  Chicago.  Described  in  School  Science 
and  Mathematics.  See  also  "Another  Algebraic  Balance,"  by 
N.  J.  Lennes,  id.,  Nov.  1905;  and  "Content-Problems  for  High 
School  Algebra,"  by  G.  W.  Meyers,  id.,  Jan.  1907,  reprinted 
from  School  Review. 

90 


POSITIVE   AND   NEGATIVE   NUMBERS.  9I 

weight  of  equal  mass  in  the  negative  scale  pan.  The 
two  weights  are  of  equal  mass,  as  the  two  numbers 
are  of  equal  arithmetical  value.  When  the  weight  is 
put  into  either  scale  pan,  it  represents  a  "real,"  or 
quality,  number ;  it  becomes  either  +  or  -. 

The  unfortunate  expression  "less  than  nothing"  (due 
to  Stifel),  the  attempt  to  consider  negative  numbers 
apart  from  positive  and  to  teach  negative  after  posi- 
tive, and  the  name  "fictitious"  for  negative  numbers, 
all  seem  absurd  enough  now ;  but  they  became  so  only 
when  the  real  significance  of  positive  and  negative  as 
opposites  was  clearly  seen.  The  value  of  the  illustra- 
tion from  debts  and  credits  (due  to  the  Hindus)  and 
from  the  thermometer,  lies  in  the  aptness  for  bringing 
out  the  oppositeness  of  positive  and  negative. 

For  the  illustration  from  directed  lines,  see  Fig.  3 
on  the  following  page. 

It  is  appropriate  that  the  advertisements  of  the  al- 
gebraic balance  use  the  quotation  from  Cajori's  His- 
tory of  Elementary  Mathematics:  "Negative  numbers 
appeared  'absurd'  or  'fictitious'  so  long  as  mathe- 
maticians had  not  hit  upon  a  visual  or  graphical  rep- 
resentation of  them .  .  .  Omit  all  illustrations  by  lines, 
or  by  the  thermometer,  and  negative  numbers  will  be 
as  absurd  to  modern  students  as  they  were  to  the  early 
algebraists." 


-o 


R 


VISUAL    REPRESENTATION    OF    COMPLEX 

NUMBERS. 

If  the  sect  OR,  one  unit  long  and  extending  to  the 
right  of  O,  be  taken  to  represent  +  1,  then  -  1  will  be 
represented  by  OL,  extending  one  to  the  left  of  O. 

+a  would  be  pictured 
U  by  a  line  a  units  long 

and  to  the  right;  -a, 
a  units  long  and  to  the 
left.  This  simplest  and 
best-known  use  of  di- 
rected lines  gives  us  a 
geometric  representa- 
tion of  real  numbers. 
The  Hindus  early 
gave  this  interpreta- 
tion to  numbers  of 
opposite  quality ;  but 
it  does  not  appear 
to  have  been  given  by  a  European  until  1629,  by 
Girard.* 

Conceiving  the  line  of  unit  length  to  be  revolved 
in  what  is  assumed  as  the  positive  direction  (counter- 
clockwise) -  1  may  be  called  the  factor  that  revolves 
from  OR  (+1)  to  OL  (-1).  Then  V-l  is  the  factor 
which,  being  used  twice,  produces  that  result;  using 

*  Albert  Girard,  Invention  Nouvelle  en  VAlgcbre,  Amster- 
dam. Perhaps  also  the  first  to  distinctly  recognize  imaginary- 
roots  of  an  equation. 

92 


D 

Fig.  3- 


REPRESENTATION  OF  COMPLEX  NUMBERS. 


93 


it  once  as  a  factor  revolving  the  line  through  one 
of  the  two  right  angles.  Then  OU  pictures  the 
number  -f  V-l.  Similarly,  since  multiplication  of  -1 
by  -  V-l  twice  produces  +1,  -  V-l  may  be  considered 
as  the  factor  which  revolves  from  OL  through  one 
right  angle  to  OD.  If  distances  to  the  right  are  called 
+,  then  distances  to  the  left  are  -,  and  +  V-l  *  b  denotes 


o 


A 


Fig.  4. 


a  line  b  units  long  and  extending  up,  and  -  V-l  ■  b  a 
line  b  units  long  extending  down.  The  geometric 
interpretation  of  the  imaginary  was  made  by  H.  Kuhn 
in  1750,  in  the  Transactions  of  the  St.  Petersburg 
Academy. 

To  represent  graphically  the  number  a  +  b  \/~-\  ( see 
Fig.  4),  we  lay  off  OA  in  the  +  direction  and  a  units 
long;  AB,  b  units  long  and  in  the  direction  indicated 
by    V-l ;  and  draw  OB.    The  directed  line  OB  repre- 


94      A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

sents  the  complex  number  a  +  b  V-l.  And  the  length 
of  OB,  V a2  +  b2,  is  the  modulus  of  a^b  V-l.  The 
geometric  interpretation  of  such  a  number  was  made 
by  Jean  Robert  Argand,  of  Geneva,  in  his  Essai,  1806. 
The  term  "modulus"  in  this  connection  was  first  used 
by  him,  in  1814. 

These  geometric  interpretations  by  Kuhn  and  Ar- 
gand, and  especially  one  made  by  Wessel,*  who  ex- 
tended the  method  to  a  representation  in  space  of 
three  dimensions,  may  be  regarded  as  precursors  of  the 
beautiful  methods  of  vector  analysis  given  to  the  world 
by  Sir  William  Rowan  Hamilton  in  1852  and  1866 
under  the  name  "quaternions." 

The  letter  i  as  symbol  for  the  unit  of  imaginary 
numbers,  V-l,  was  suggested  by  Euler.  It  remained 
for  Gauss  to  popularize  the  sign  i  and  the  geometric 
interpretations  made  by  Kuhn  and  Argand. 

The  contrasting  terms  "real"  and  "imaginary"  as 
applied  to  the  roots  of  an  equation  were  first  used  by 
Descartes.  The  name  "imaginary"  was  so  well  started 
that  it  still  persists,  and  seems  likely  to  do  so,  although 
it  has  long  been  seen  to  be  a  misnomer. f  A  few 
writers  use  the  terms  scalar  and  orthotomic  in  place 
of  real  and  imaginary. 

The  historical  development  of  this  subject  furnishes 
an  illustration  of  the  general  rule,  that,  as  we  advance, 
each  new  generalization  includes  as  special  cases  what 
we  have  previously  known  on  the  subject.  The  gen- 
eral form  of  the  complex  number,  a  +  bi,  includes  as 
special  cases  the  real  number  and  the  imaginary.     If 

*  To  the  Copenhagen  Academy  of  Sciences,  1797. 

t  It  is  interesting  to  notice  the  prestige  of  Descartes's  usage 
in  fixing  the  language  of  algebra :  the  first  letters  of  the  alpha- 
bet for  knowns,  the  last  letters  for  unknowns,  the  present 
form  of  exponents,  the  dot  between  factors  for  multiplication. 


REPRESENTATION  OF  COMPLEX  NUMBERS.  95 

b  =  0,  a  +  bi  is  real.  If  a  =  0,  a  +  bi  is  imaginary. 
The  common  form  of  a  complex  number  is  the  sum 
of  a  real  number  and  an  imaginary.* 

In  1799  Gauss  published  the  first  of  his  three  proofs 
that  every  algebraic  equation  has  a  root  of  the  form 
a  +  bi. 

The  linear  equation  forces  us  to  the  consideration 
of  numbers  of  opposite  quality:  x-a  =  0  and  x  +  a  =  0, 
satisfied  by  the  values  -f  a  and  -  a  respectively.  The 
pure  quadratic  gives  imaginary  in  contrast  with  real 
roots:  ,r2-a2  =  0  and  x2  +  a2  =  0  satisfied  by±a  and 
ztai.    The  complete  quadratic 

ax2  +  bx  +  c  =  0 
has  for  its  roots  a  pair  of  conjugate  complex  numbers 
when  the  discriminant,  b2  -  4ac,  is  negative  and  b  is 
not  =  0. 

But  though  the  recognition  of  imaginary  and  com- 
plex numbers  is  a  necessary  consequence  of  simple 
algebraic  analysis,  no  complete  understanding  or  ap- 
preciation of  them  is  possible  until  there  is  some  tan- 
gible or  visible  representation  of  them.  History's 
lesson  to  us  in  this  respect  is  plain :  positive  and  nega- 
tive, imaginary,  and  complex  numbers  must  be  graph- 
ically represented  in  teaching  algebra. 

The  algebraic  balance  mentioned  on  page  90  might 
be  further  developed  by  the  addition  of  an  appliance 
whereby  imaginary  numbers  should  be  illustrated,  a 
weight  put  into  a  certain  pan  having  the  effect  of 
pulling  the  main  beam  to  one  side,  and  arrangements 
for  pulling  the  beam  in  several  other  directions  to  illus- 
trate complex  numbers. 

*  Professor  Schubert  (p.  24)  adds  that  "we  have  found  the 
most  general  numerical  form  to  which  the  laws  of  arithmetic 
can  lead." 


96       A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

If  in  a  football  game  we  denote  the  forces  exerted 
in  the  direction  OR  (in  Fig.  3)  by  positive  real  num- 
bers, then  the  opponents'  energy  exerted  in  exactly 
the  opposite  direction,  OL,  will  be  denoted  by  negative 
numbers.  Forces  in  the  line  of  OU  or  OD  will  be 
denoted  by  imaginary  numbers;  and  all  other  forces 
in  the  game,  acting  in  any  other  direction  on  the 
field,  will  be  denoted  by  complex  numbers  of  the  gen- 
eral type.* 

Each  force  represented  by  a  general  complex  num- 
ber is  resolvable  into  two  forces,  one  represented  by 
a  real  number  and  the  other  by  an  imaginary,  as  OB 
(in  Fig.  4)  is  the  resultant  of  OA  and  AB. 

A  trigonometric  representation  of  an  imaginary  num- 
ber as  exponent  is  furnished  by  the  formula 

e*  =  cos  1  +  i  sin  1 . 

*  Illustration  from  Taylor's  Elements  of  Algebra,  where  the 
visual  representation  of  imaginary  and  complex  numbers  is 
made  in  full. 


ILLUSTRATIONS  OF  THE  LAW  OF  SIGNS  IN 
ALGEBRAIC  MULTIPLICATION, 

A  Geometric  Illustration. 

If  distances  to  the  right  of  O  be  called  +,  then  dis- 
tances to  the  left  will  be  -.  Call  distances  up  from 
O  +,  and  those  down  -.  Rectangle  OR  has  ab  units 
of  area.    Assume  that  the  product  ab  is  +. 


R' 

T 

R 

■f 

b 

S' 

— a 

+a 

S 

0 

- 

b* 

Fig.  5- 


Suppose  SR  to  move  to  the  left  until  it  is  a  units 
to  the  left  of  O,  in  the  position  S'R'.  The  base  di- 
minished, became  zero,  and  passed  through  that  value, 
and  therefore  is  now  negative ;  so  also  the  rectangle. 
The  product  of  -a  and  +b  is  -ab. 

97 


98       A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

Suppose  TR'  to  move  downward  until  it  is  b  units 
below  O.  The  rectangle,  previously  -,  has  passed 
through  zero,  and  must  now  be  +.  The  product  of 
-a  and  -b  is  +ab. 

Similarly    (+a)  (~b)=-ab. 

From  a  Definition   of  Multiplication. 

Multiplication  is  the  process  of  performing  upon 
one  of  two  given  numbers  (the  multiplicand)  the 
same  operation  which  is  performed  upon  the  primary 
unit  to  obtain  the  other  number   (the  multiplier.)* 

When  the  multiplier  is  an  arithmetical  integer,  the 
primary  unit  is  that  of  arithmetic,  1,  and  we  have  the 
special  case  that  is  correctly  defined  in  the  primary 
school  as,  "taking  one  number  as  many  times  as  there 
are  units  in  another." 

Suppose  we  are  to  multiply  +4  by  +3.  Assuming 
+1  as  the  primary  unit,  the  multiplier  is  produced  by 

*  In  this  definition,  "the  same  operation  which  is  performed 
upon  the  primary  unit  to  obtain  the  multiplier"  is  to  be  under- 
stood to  mean  the  most  fundamental  operation  by  which  the 
multiplier  may  be  obtained  from  unity,  or  that  operation  which 
is  primarily  signified  by  the  multiplier.  E.  g.,  If  the  multi- 
plier is  2,  this  number  primarily  means  unity  taken  twice,  or 
the  unit  added  to  itself;  multiplying  4  by  2  therefore  means 
adding  4  to  itself,  giving  the  result  8.  Dr.  Young,  in  his  new 
book,  The  Teaching  of  Mathematics,  p.  227,  says  that  as  2  is 
i  +  i2,  therefore  2X4  would  by  this  definition  be  4+42,  or  20; 
or,  as  2  is  i-f  1/1,  therefore  2X4  would  be  4+4/4,  or  5;  etc. 
But  while  it  is  true  that  i  +  i2  and  i-f-1/1  are  each  equal  to  2, 
neither  of  them  is  the  primary  signification  of  2,  or  represents 
2  in  the  sense  of  the  definition.  Neither  of  them,  is  a  proper 
statement  of  the  multiplier  "within  the  meaning  of  the  law." 

It  is  not  maintained  that  this  definition  has  no  difficulties, 
or  that  it  directly  helps  a  learner  in  comprehending  the  mean- 
ing of  such  a  multiplication  as  V2XV3",  but  only  that  it  is  a 
generalization  that  is  helpful  for  the  purpose  for  which  it  is 
used,  and  that  it  is  in  line  with  the  fundamental  idea  of  multi- 
plication so  far  as  that  idea  is  understood. 

The  definition  is  only  tentative,  and  this  treatment  does  not 
pretend  to  be  a  proof. 


THE  LAW  OF  SIGXS.  (JO, 

taking  that  unit  additively  "three  times/'  (+l)+(+l) 
+(+1).  That  is  what  the  number  +3  means;  and  to 
multiply  +4  by  it,  means  to  do  that  to  +4.  (+4)+(+4) 
+(+4)  =+ 12.  Similarly,  the  product  of  -4  by  +3  = 
(_4)  +  (-4)  +  (-4)  =  -12. 

To  multiply  +4  by  -3 :  The  multiplier  is  the  result 
obtained  by  taking  three  times  additivelv  the  primary 
unit  with  its  quality  changed.  The  product  of  +4  by 
-3  is  therefore  the  result  obtained  by  taking  three 
times  additively  +4  with  its  quality  changed.  (-4)  + 
(-4)  +  (-4)  =-12.  Similarly,  to  multiply  -4  by  -3 
is  to  take  three  times  additively  -4  with  its  quality 
changed:   (+4)+(+4)+(+4)  =+12. 

Summarizing  the  four  cases,  we  have  "the  law  of 
signs" :  the  product  is  +  when  the  factors  are  of  like 
quality,  -  when  they  are  of  unlike  quality. 

A  more  General  Form  of  the  Lazu  of  Signs. 

In  deriving  the  law  from  the  definition  of  multi- 
plication, the  primary  unit  was  assumed  as  +1.  Assume 
-1  as  the  primary  unit,  and  multiply  +4  by  +3.  The 
multiplier,  +3,  is  obtained  from  the  primary  unit,  -1, 
by  taking  three  times  additively  the  unit  with  its  sign 
changed.  Performing  the  same  operation  on  the  mul- 
tiplicand, +4,  we  have  (-4)  +  (-4)  +  (-4)  =-12.  Simi- 
larly, the  product  of  -4  by  +3  =  (+4)  +  (+4)  +  (+4)  = 
+  12.  To  multiply  +  4  by  -  3  :  The  multiplier  is  the  re- 
sult obtained  by  taking  three  times  additively  the  unit, 
-  1,  without  change  of  sign  ;  therefore  the  product  of 
+  4  by  -3  =(+4)  +  (+4)  +  (+4)  =+12.  So  also  -4 
multiplied  by  -3  gives  -  12.  Summarizing  these  four 
cases,  we  have  the  law  of  signs  when  -  1  is  taken  as 
the  primary  unit :  the  product  is  -  when  the  factors 
are  of  like  quality,  +  when  they  are  of  unlike  quality. 


IOO    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

In  the  geometric  illustration  above,  we  first  assumed 
the  rectangle  +a  by  +b  to  be  +.  Assuming  the  contrary, 
the  sign  of  each  subsequent  product  is  reversed,  and 
we  have  a  geometric  illustration  of  the  law  of  signs 
when  -1  is  taken  as  the  primary  unit. 

The  law  of  signs  taking  +1  as  the  primary  unit,  and 
that  taking  -1  as  the  primary  unit,  may  be  combined 
into  one  law  thus :  If  the  two  factors  are  alike  in  qual- 
ity, the  product  is  like  the  primary  unit  in  quality;  if 
the  two  factors  are  opposite  in  quality,  the  quality  of 
the  product  is  opposite  to  that  of  the  primary  unit. 
Or:  Like  signs  give  like  (like  the  primary  unit)  ;  un- 
like signs  give  unlike  (the  unit). 

The  assumption  of  still  other  numbers  as  primary 
unit  leads  to  other  laws — other  "algebras." 

Multiplication  as  a  Proportion. 

Since  by  definition  a  product  bears  the  same  relation 
to  the  multiplicand  that  the  multiplier  bears  to  the 
primary  unit,  this  equality  of  relation  may  be  stated 
in  the  form  of  a  proportion : 

product  :  multiplicand   : :  multiplier  :  primary  unit 
or, 

primary  unit  ':  multiplier  : :  multiplicand  :  product. 

Gradual  Generalization  of  Multiplication. 

From  the  time  when  Pacioli  found  it  necessary  (and 
difficult)  to  explain  how,  in  the  case  of  proper  frac- 
tions in  arithmetic,  the  product  is  less  than  the  multi- 
plicand, to  the  present  with  its  use  of  the  term  multi- 
plication in  higher  mathematics,  is  a  long  evolution. 
It  is  one  of  the  best  illustrations  of  the  generalization 
of  a  term  that  was  etymologically  restricted  at  the 
beginning. 


EXPONENTS. 

The  definition  of  exponent  found  in  the  elementary 
algebras  is  sufficient  for  the  case  to  which  it  is  applied 
— the  case  in  which  the  exponents  are  arithmetic  in- 
tegers. Our  assumption  of  a  primary  unit  for  algebra 
being  what  it  is,  the  distinction  between  arithmetic 
numbers  as  exponents  and  positive  numbers  as  ex- 
ponents is  usually  neglected.  Or  we  may  simply  define 
pos'tive  exponent.  The  meaning  of  negative  arid  frac- 
tional exponents  is  easily  deduced.  In  fact  those  who 
first  used  exponents  and  invented  an  exponential  nota- 
tion (Oresme  in  the  fourteenth  century  and  Stevin  in- 
dependently in  the  sixteenth)  had  fractions  as  well  as 
whole  numbers  as  exponents.  And  negative  exponents 
had  been  invented  before  Wallis  studied  them  in  the 
seventeenth  century.  Each  of  these  can  be  defined 
separately.  And  modern  mathematics  has  used  other 
forms  of  exponents.  They  have  been  made  to  follow 
the  laws  of  exponents  first  proved  for  ordinary  in- 
tegral exponents,  and  their  significance  has  been  as- 
signed in  conformity  thereto.  Each  separate  species 
of  exponent  is  defined.  A  unifying  .conception  of  them 
all  might  express  itself  in  a  definition  covering  all 
known  forms  as  special  cases.  The  general  treatment 
of  exponents  is  yet  to  come. 

Wanted:  a  definition  of  exponent  that  shall  be 
general  for  elementary  mathematics. 


C      «  *  €     C 


AN  EXPONENTIAL  EQUATION. 

The  chain-letters,  once  so  numerous,  are  now — it 
is  to  be  hoped — obsolete.  In  the  form  that  was  prob- 
ably most  common,  the  first  writer  sends  three  letters, 
each  numbered  1.  Each  recipient  is  to  copy  and  send 
three,  numbered  2,  and  so  on  until  number  SO  is 
reached. 

Query :  If  every  one  were  to  do  as  requested,  and  it 
were  possible  to  avoid  sending  to  any  person  twice, 
what  number  of  letter  would  be  reached  when  every 
man,  woman  and  child  in  the  world  should  have  re- 
ceived a  letter? 

Let  n  represent  the  number.  Take  the  population 
of  the  earth  to  be  fifteen  hundred  million.  Then  this 
large  number  is  the  sum  of  the  series 

3,  32,  33...3* 
„     a  (rn-l)      3  (3"-l) 


r-1  2 

|(3"-1)  =1,500,000,000 

3W-1  =  1,000,000,000 

n\og  3  =  log  (109) 

n  =  z-9'-z  =  18.86 
log  3 

There  are  not  enough  people  in  the  world  for  the 
letters  numbered  19  to  be  all  sent. 


TWO    NEGATIVE    CONCLUSIONS    REACHED 
IN  THE  NINETEENTH  CENTURY. 

1.  That  general  equations  above  the  fourth  degree 
are  insoluble  by  pure  algebra. 

The  solution  of  equations  of  the  third  and  fourth 
degree  had  been  known  since  1545.  Two  centuries 
and  a  half  later,  young  Gauss,  in  his  thesis  for  the 
doctorate,  proved  that  every  algebraic  equation  has 
a  root,  real  or  imaginary.*  He  made  the  conjecture 
in  1801  that  it  might  be  impossible  to  solve  by  radicals 
any  general  equation  of  higher  degree  than  the  fourth. 
This  was  proved  by  Abel,  a  Norwegian,  whose  proof 
was  printed  in  1824,  when  he  was  about  twenty-two 
years  old.  Two  years  later  the  proof  was  published'  in 
an  expanded  form,  with  more  detail. 

Thus  inventive  effort  was  turned  in  other  directions. 

2.  That  the  "parallel  postulate"  of  Euclid  can  never 
be  proved  from  the  other  postulates  and  axioms. 

Ever  since  Ptolemy,  in  the  second  century,  the  at- 
tempt had  been  made  to  prove  this  postulate,  or 
''axiom,"  and  thus  place  it  among  theorems.     In  1826, 

*  Of  this  proof,  published  when  Gauss  was  twenty-two  years 
old,  Professor  Maxime  Bocher  remarks  (Bulletin  of  Amer. 
Mathematical  Society,  Dec.  1904,  p.  118,  note)  :  "Gauss's  first 
proof  (1799)  that  every  algebraic  equation  has  a  root  gives  a 
striking  example  of  the  use  of  intuition  in  what  was  intended 
as  an  absolutely  rigorous  proof  by  one  of  the  greatest  and  at 
the  same  time  most  critical  mathematical  minds  the  world  has 
ever  seen."  It  should  be  added  that  Gauss  afterward  gave 
two  other  proofs  of  the  theorem. 

103 


104    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

Lobachevskv,  professor  and  rector  at  the  University 
of  Kasan,  Russia,  proved  the  futility  of  the  attempt, 
and  published  his  proof  in  1829.  He  constructed  a 
self-consistent  geometry  in  which  the  other  postulates 
and  axioms  are  assumed  and  the  contrary  of  this,  thus 
showing  that  this  is  independent  of  them  and  therefore 
can  not  be  proved  from  them.  No  notice  of  his  re- 
searches appeared  in  Germany  till  1840.  In  1891 
Lobachevsky's  work  was  made  easily  available  to  Eng- 
lish readers  through  a  translation  by  Prof.  George 
Bruce  Halsted.* 

The  effort  previously  expended  in  attempting  the 
impossible  was  henceforth  to  be  turned  to  the  develop- 
ment of  non-Euclidean  geometry,  to  investigating  the 
consequences  of  assuming  the  contrary  of  certain  ax- 
ioms, to  w-dimensional  geometry.  "As  is  usual  in 
every  marked  intellectual  advance,  every  existing  diffi- 
culty removed  has  opened  up  new  fields  of  research, 
new  tendencies  of  thought  and  methods  of  investiga- 
tion, and  consequently  new  and  more  difficult  problems 
calling  for  solution. "f 

High-school  geometry  must  simply  assume  (choose) 
Euclid's  postulate  of  parallels,  perhaps  preferably  in 
Playfair's  form  of  it:  Two  intersecting  lines  can  not 
both  be  parallel  to  the  same  line. 

*  Austin,  Texas,  1892.  It  contains  a  most  interesting  intro- 
duction by  the  translator.  Dr.  Halsted  translated  also  Bolyai's 
work  (1891),  compiled  a  Bibliography  of  Hyperspace  and 
Non-Euclidean  Geometry  (1878)  of  174  titles  by  81  authors, 
and  has  himself  written  extensively  on  the  subject,  being 
probably  the  foremost  writer  in  America  on  non-Euclidean 
geometry  and  allied  topics. 

f  Withers,  p.  63-4. 


THE  THREE  PARALLEL  POSTULATES 
ILLUSTRATED. 

In  contrast  to  Euclid's  postulate  (just  quoted)  Loba- 
chevsky's  is,  that  through  a  given  point  an  indefinite 
number  of  lines  can  be  drawn  in  a  plane,  none  of  which 
cut  a  given  line  in  the  plane,  while  Riemann's  postu- 
late is,  that  through  the  point  no  line  can  be  drawn 
in  the  plane  that  will  not  cut  the  given  line.  Thus  we 
have  three  elementary  plane  geometries. 

An  excellent  simple  illustration  of  the  contrast  has 
been  devised :  Let  AB  and  PC  be  two  straight  lines  in 


Fig.  6. 

the  same  plane,  both  unlimited  in  both  directions ;  AB 
fixed  in  position ;  and  PC  rotating  about  the  point  P, 
say  in  the  positive  (counter-clockwise)  direction,  inter- 
secting first  toward  the  right  as  shown  in  Figure  6. 

'Three  different  results  are  logically  possible.  When 
the  rotating  line  ceases  to  intersect  the  fixed  line  in 
one  direction    [toward  the  right]    it  will  immediately 


106    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

intersect  in  the  opposite  direction  [toward  the  left], 
or  it  will  continue  to  rotate  for  a  time  before  inter- 
section takes  place,  or  else  there  will  be  a  period  of 
time  during  which  the  two  lines  intersect  in  both  direc- 
tions. The  first  of  these  possibilities  gives  Euclid's, 
the  second  Lobachevsky's,  and  the  third  Riemann's 
geometry. 

"The  mind's  attitude  toward  these  three  possibilities 
taken  successively  illustrates  in  a  curious  way  the 
essentially  empirical  nature  of  the  straight  line  as  we 
conceive  it.  Logically  one  of  these  possibilties  is  just 
as  acceptable  as  the  other.  From  this  point  of  view 
strictly  taken  there  is  certainly  no  reason  for  pre- 
ferring one  of  them  to  another.  Psychologically,  how- 
ever, Riemann's  hypothesis  seems  absolutely  contra- 
dictory, and  even  Euclid's  is  not  quite  so  acceptable 
as  that  of  Lobachevsky." 

As  a  slight  test  of  the  relative  acceptability  of  these 
hypotheses  to  the  unsophisticated  mind,  the  present 
writer  drew  on  the  blackboard  a  figure  like  that  above, 
mentioned  in  simple  language  the  three  possibilities, 
and  asked  pupils  to  express  opinion  on  slips  of  paper. 
Forty-six  out  of  54  voted  that  the  second  is  the  true 
one.  Two  said  they  "guessed"  it  is,'  twenty -one 
"thought"  so,  thirteen  "felt  sure,"  and  ten  "knew." 
Six  "thought"  that  the  first  supposition  is  correct,  and 
two  "felt  sure"  of  it.  No  one  voted  for  the  third,  and 
the  writer  has  never  heard  but  one  person  express 
opinion  in  favor  of  the  third  supposition.  Some  of 
the  pupils  had  had  a  few  weeks  of  plane  geometry. 
Of  these,  most  who  voted  in  the  majority  wanted  to 
change  as  soon  as  it  was  pointed  out  that  the  second 
supposition  implies  that  two  intersecting  lines  can 
both  be  parallel  to  the  same  line.     Undoubtedly  some 


THE  THREE  PARALLEL  POSTULATES.       I07 

of  the  more  immature  were  unable  to  grasp  the  idea 
that  the  lines  are  of  unlimited  length,  and  possibly  it 
may  be  somewhat  general  that  those  who  favor  the 
second  supposition  do  not  fully  grasp  that  idea.  Such 
a  test  merely  illustrates  that  Euclid's  postulate  is  not 
in  all  its  forms  apodictic. 

The  whole  question  of  parallel  postulates  is  ad- 
mirably treated  by  Dr.  Withers/1'  to  whose  book  (p. 
117)  the  writer  is  indebted  for  the  two  paragraphs 
quoted  above. 

In  trigonometry.  The  familiar  figure  in  trigonom- 
etry representing  the  line  values  of  the  tangent  of  an 
angle  at  the  center  of  a  unit  circle  as  the  angle  in- 
creases and  passes  through  90°  is  another  form  of 
this  figure.  And  the  assumption  that  intersection  of 
the  final  (revolving)  side  with  the  line  of  tangents 
begins  at  an  infinite  distance  below  at  the  instant  it 
ceases  above,  places  our  trigonometry  on  a  Euclidean 
basis. 

Parallels  meet  at  infinity.  Kepler's  definition  would 
seem  paradoxical  if  offered  in  elementary  geometry, 
but  is  valuable  in  more  advanced  work,  and  is  intelli- 
gible enough  when  made  in  the  language  of  limits. 
Let  PP'  be  perpendicular  to  SQ ;  let  Q  move  farther 
and  farther  to  the  right  while  P  remains  fixed;  and 
let  PTR  be  the  limit  toward  which  angle  P'PQ  ap- 
proaches as  the  distance  of  Q  from  P'  increases  with- 


*John  William  Withers,  Euclid's  Parallel  Postulate:  Its 
Nature,  Validity,  and  Place  in  Geometrical  Systems,  his  thesis 
for  the  doctorate  at  Yale,  published  by  The  Open  Court  Pub- 
lishing Co.,  1905.  It  includes  a  bibliography  of  about  140 
titles  on  this  and  more  or  less  closely  related  subjects,  men- 
tioning Halsted's  bibliography  of  174  titles  and  Roberto 
Bonola's  of  353  titles.  To  these  lists  might  be  added  Man- 
ning's Non-Euclidean  Geometry  (1901)  which  is  brief,  ele- 
mentary and  interesting. 


108    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

out  limit.*  Then  PR  is  parallel  to  SQ.  That  is,  paral- 
lelism is  attributed  to  the  limiting  position  of  inter- 
secting lines  as  the  point  of  intersection  recedes  with- 
out limit ;  which,  for  the  sake  of  brevity,  we  may  ex- 
press by  the  familiar  sentence,  ''Parallels  meet  at  in- 
finity." 

The  three  postulates  again.  Now  suppose  PS  to 
move,  P  remaining  fixed  and  S  moving  to  the  left, 
TPP'  being  the  limit  of  angle  SPP'  as  P'S  increases 
without  limt.    Then  PT  is  parallel  to  SQ.    According 


^S 


P' 


Fig.  7- 


to  Euclid's  postulate  PT  and  PR  are  one  straight  line ; 
according  to  Lobachevsky's  they  are  not;  while  ac- 
cording to  Riemann's  Q  and  S  can  not  recede  to  an 
infinite  distance  (but  Q  comes  around,  so  to  speak, 
through  S,  to  P'  again)  and  there  is  no  limiting  posi- 
tion (in  the  terminology  of  the  theory  of  limits)  and 
no  parallel  in  the  Euclidean  sense  of  the  term. 

*  In  Fig.  6  the  moving  line  rotated  until  after  it  ceased  to 
intersect  the  fixed  line  toward  the  right.  In  the  present  illus- 
tration (Fig.  7)  PQ  rotates  only  as  Q,  the  point  of  intersec- 
tion, recedes  along  the  line  SP'Q. 


GEOMETRIC  PUZZLES. 

*"A  rectangular  hole  13  inches  long  and  5  inches 
wide  was  discovered  in  the  bottom  of  a  ship.  The 
ship's  carpenter  had  only  one  piece  of  board  with 
which  to  make  repairs,  and  that  was  but  8  inches 
square    (64  square  inches)    while  the  hole  contained 


Fig.  8. 


65  square  inches.  But  he  knew  how  to  cut  the  board 
so  as  to  make  it  fill  the  hole* !  Or  in  more  prosaic 
form : 

Fig.  8  is  a  square  8  units  on  a  side,  area  64;  cut' it 

109 


IIO    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 


through  the  heavy  lines  and  rearrange  the  pieces  as 
indicated  by  the  letters  in  Fig.  9,  and  you  have  a  rect- 
angle 5,  by  13v  area  65.     Explain.         . 


i     1 — 1 — % 

1         '                      / 

1        '         ■         '/ 
T Tr~  I J—' 

/' 

» —  1-/^-  1 

1                     // 

'" 

__)'_/_bJ!___ 

.-  J 

-     -      J 

L             //            '                 ' 

#H    '     I 

1  /7    '        '         ' 

/       »        iD     I 

ff-|-  x_a__ 

/    »       i       *'       i 
V— >- — 1 L_     ! 

Fig.  9. 


Fig.  io. 


Fig.  10  explains.  EH  is  a  straight  line,  and  HG 
is  a  straight  line,  but  they  are  not  parts  of  the  same 
straight  line.     Proof: 

Let  X  be  the  point  at  which  EH  produced  meets 


GEOMETRIC   TL'ZZLES.  Ill 

GJ ;  then  from  the  similarity  of  triangles  EHK  and 
EXJ 

XJ  :HK  =  EJ  :EK 
XJ  :3=13  :8 
XJ  =  4.875 
But  GJ  =  5. 

Similarly,   EFG  is  a  broken  line. 

The  area  of  the  rectangle  is,  indeed,  65,  but  the  area 
of  the  rhomboid  EFG-H  is  1. 

Professor  Ball*  uses  this  to  illustrate  that  proofs  by 
dissection  and  superposition  are  to  be  regarded  with 
suspicion  until  supplemented  by  mathematical  reason- 
ing. 

'This  geometrical  paradox...  seems  to  have  been 
well  known  in  1868,  as  it  was  published  that  year  in 
Schlomilclrs  Zeitschrift  fiir  Mathematik  und  Physik, 
Vol.  13,  p.  162." 

In  an  article  in  The  Open  Court,  August  1907,  (from 
which  the  preceding  four  lines  are  quoted),  Mr.  Escott 
generalizes  this  puzzle.  The  puzzle  is  so  famous  that 
his  analysis  can  not  but  be  of  interest.  With  his  per- 
mission it  is  here  reproduced : 

In  Fig.  11,  it  is  shown  how  we  can  arrange  the  same 
pieces  so  as  to  form  the  three  figures,  A,  B,  and  C. 
If  we  take  x  =  5,  v  =  3,  we  shall  have  A  =  63,  B  =  64, 
C  =  65. 

Let  us  investigate  the  three  figures  by  algebra. 

A  =  2xy  +  2xy  +  y(2y-x)  =  3xy  +  2y2 

B  =  (x  +  y) -  -  x2  +  2xy  +  y2 

C  =  x(2x  +  y)  =  2x2  +  xy 
C  -  B  =  x2  -  xy  -  y2 
B  -  A  =  x2  -  xy  -  y2. 

*  Recreations,  p.  49. 


X 

y 

x-fy 

^^ 

y            ^"^ 

X 

y 

X 

x+y 

B 


V 

x 

y 

X 

X 

x+y               ^*^ 

X 

Fig.  ii, 

112 


GEOMETRIC  PUZZLES.  113 

These  three  figures  would  be  equal  if  x2  -xy-y2 
=  0,  i.  e.,  if 

x    1  +  a/5 
y~      2 
so  the  three  figures  cannot  be  made  equal  if  x  and  y 
are  expressed  in  rational  numbers. 

We  will  try  to  find  rational  values  of  x  and  y  which 
will  make  the  difference  between  A  and  B  or  between 
B  and  C  unity. 

Solving  the  equation 

x2  -  xy  -  y2  =  ±  1 
we  find  by  the  Theory  of  Numbers  that  the  y  and  x 
may  be  taken  as  any  two  consecutive  numbers  in  the 
series 

1,  2,  3,  5,  8,  13,  21,  34,  55, 

where  each  number  is  the  sum  of  the  two  preceding 
numbers. 

The  values  y  =  3  and  x  -  5  are  the  ones  commonly 
given.   For  these  we  have,  as  stated  above,  A  <  B  <  C. 

The  next  pair,  x  =  8,  y  =  5  give  A  >  B  >  C,  i.  e., 
A  =  170,  B  =  169,  C=168. 

Fig.  12  shows  an  interesting  modification  of  the 
puzzle. 

A  =  4/rv  +  (y  +  x)  (2y  -  x)  =  2y2  +  2yz  +  3xy  -  xz 

B  =  (x  +  y  +  z) 2  -  x2  +  y2  +  £2  +  2y^  +  2^i-  +  2xy 

C  =  (x  +  2z)  (2x  +  y  +  z)  =  2x2  -f  2z2  +  2y-  +  5-srjr  4-  xy 

When  at  =  6,  j  =-5>,*  =  1  we  have  A  =  B  =  C  =  144. 

When  x  =  10,  y  =  10,  z  -  3  we  have  A>  B  >  C,  viz., 
A  =  530,  B  =  529,  C  =  528. 

Another  puzzle  is  made  by  constructing  a  cardboard 
rectangle  13  by  11,  cutting  it  through  one  of  the 
diagonals  (Fig.  13)  and  sliding  one  triangle  against 
the   other  along    their   common    hypotenuse    to    the 


B 


z\           y 

y      \z 

X 

x+z           y 

x  +  z 

y 

*+• 

y  +  z 

z 

x+z 

X 

Fig.  12. 


M 


GEOMETRIC  PUZZLES. 


115 


position  shown  in  Fig.  14.  Query:  How  can  Fig.  14 
be  made  up  of  square  VRXS,  12  units  on  a  side,  area 
144,  +  triangle  PQR,  area  0.5,  +  triangle  STU,  area 
0.5,  -total  area  145  ;  when  the  area  of  Fig.  13  is  only 
143? 

Inspection  of  the  figures,  especially  if  aided  by  the 
cross  lines,  will  show  that  VRXS  is  not  a  square.  VS 
is  12  long;  but  SX  <  12.  TX=11  (the  shorter  side 
in  Fig.  13)  but  ST  <  1   (see  ST  in  Fig.'  13). 


!   !   •   !   !    !   !    :   l 

__,_            -     -               1 1 k/-     ►.           ' 

1       s'      y 
1         1  / 

_  J  .  JfT  _  - 

Vl"  "' — 

1 

I 

1      •      1      >      1      1  A           ' 

--  ^  -  -;  -  -:--;--  h  -^K^-  -  -  -  4  -  - 

1     1     1    yf 1     I      ' 

-     1      -1  - 
1        1 

1          1 
1          1          | 
l          1          . 
1          , 

,        *     X    'it'll 
/!       ,       1              iii', 

/      1         _.            1            \            1            1            1           X            l 

Fig.  13. 

ST  :  VP  =SU  :VU 

ST  :  11  =  1  :  13 
ST  -  iy 

Rectangle  VRXS  =  12  x  1 1"/18  =  1422/13 

Triangle  PQR  =  triangle  STU  -  */,  ■  n/13  *  1  =  "/» 
Fig.  14  =  rectangle  +  2  triangles 
=  142V„  +  "As  =  143- 


Il6    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 


By  sliding  the  triangles  one  place  (to  the  first  cross 
line)  in  the  other  direction  we  appear  to  have  a  rect- 
angle 14  by  10  and  two  small  triangles  with  an  area 
of  y2  each,  total  area  141 — as  much  smaller  than  Fig. 
13  as  Fig.  14  is  larger.  Slide  the  triangles  one  more 
place  in  the  direction  last  used,  and  the  apparent  area 


Fig.  14. 

is  139.  The  explanation  is  of  course  similar  to  that 
given  for  Fig.  14. 

This  paradox  also  might  be  treated  by  an  analysis 
resembling  that  by  which  Mr.  Escott  has  treated  the 
preceding. 

Very  similar  is  a  puzzle  due  to  S.  Loyd,  "the 
puzzlist."  Fig.  A  is  a  square  8x8,  area  64  Fig.  B 
shows  the  pieces  rearranged  in  a  rectangle  apparently 
7x9,  area  63. 


GEOMETRIC  PUZZLES. 


II7 


Paradromic  rings*  A  puzzle  of  a  very  different 
sort  is  made  as  follows.  Take  a  strip  of  paper,  say 
half  as  wide  and  twice  as  long  as  this  page ;  give  one 
end  a  half  turn  and  paste  it  to  the  other  end.  The  ring 
thus  formed  is  used  in  theory  of  functions  to  illustrate 
a  surface  that  has  only  one  face :  a  line  can  be  drawn 
on  the  paper  from  any  point  of  it  to  any  other  point 


iX^^ 

8 

7 

7 

8 

1 

Fig.  A. 


of  it,  whether  the  two  points  were  on  the  same  side 
or  on  opposite  sides  of  the  strip  from  which  the  ring 
was  made.  The  ring  is  to  be  slit — cut  lengthwise  all 
the  way  around,  making  the  strip  of  half  the  present 
width.  State  in  advance  what  will  result.  Try  and 
see.  Now  predict  the  effect  of  a  second  and  a  third 
slitting. 


*  The   theory  of  these   rings   is   due  to   Listing,    Topologie, 
part  10.     See  Ball's  Recreations,  p.  75-6. 


DIVISION  OF  PLANE  INTO  REGULAR  POLY- 
GONS. 

The  theorem  seems  to  have  been  pleasing  to  the 
ancients,  as  it  is  to  high-school  pupils  to-day,  that  a 
plane  surface  can  be  divided  into  equilateral  triangles, 
squares,  or  regular  hexagons,  and  that  these  are  the 
only  regular  polygons  into  which  the  surface  is  divis- 
ible.   As  a  regular  hexagon  is  divided  by  its  radii  into 


Fig.  15. 


Fig.  16. 


six  equilateral  triangles,  the  division  of  the  surface 
into  triangles  and  hexagons  gives  the  same  arrange- 
ment (Fig.  16). 

The  hexagonal  form  of  the  bee's  cell  has  long  at- 
tracted attention  and  admiration.  The  little  worker 
could  not  have  chosen  a  better  form  if  he  had  had  the 
advantage  of  a  full  course  in  Euclid !  The  hexagon  is 
best  adapted  to  the  purpose.     It  was  discussed  from  a 

118 


REGULAR   POLYGONS.  II9 

mathematical  point  of  view  by  Maclaurin  in  one  of  the 
last  papers  he  wrote.*  It  has  been  pointed  outf  that 
the  hexagonal  structure  need  not  be  attributed  to  mech- 
anical instinct,  but  may  be  due  solely  to  external  pres- 
sure. (The  cells  of  the  human  body,  originally  round, 
become  hexagonal  under  pressure  from  morbid 
growth.) 

Agricultural  journals  are  advising  the  planting  of 
trees  (as  also  corn  etc.)  on  the  plan  of  the  equilateral 
triangle  instead  of  the  square.  Each  tree  is  as  far 
from  its  nearest  neighbors  in  Fig.  16  as  in  Fig.  15. 
The  circles  indicated  in  the  corner  of  each  figure 
represent  the  soil  etc.  on  which  each  tree  may  be  sup- 
posed to  draw.  The  circles  in  Fig.  16  are  as  large 
as  in  Fig.  15  but  there  is  not  so  much  space  lost  be- 
tween them.  As  the  distance  from  row  to  row  in 
Fig.  16  =  the  altitude  of  one  of  the  equilateral  triangles 
-  2  V3  =  0.866  of  the  distance  between  trees,  it  re- 
quires (beyond  the  first  row)  only  87%  as  much 
ground  to  set  out  a  given  number  of  trees  on  this 
plan  as  is  required  to  set  them  out  on  the  plan  of  Fig. 
15.  It  may  be  predicted  that,  as  land  becomes  scarce, 
pressure  will  force  the  orchards,  gardens  and  fields 
into  a  uniformly  hexagonal  arrangement. 

*  In  Philosophical  Transactions  for  1743. 

f  See  for  example  E.  P.  Evans's  Evolutional  Ethics  and 
Animal  Psychology,  p.  205. 


A  HOMEMADE   LEVELING   DEVICE. 

The  newspapers  have  been  printing  instructions  for 
making  a  simple  instrument  useful  in  laying  out  the 
grades  for  ditches  on  a  farm,  or  in  simlar  work  in 
which  a  high  degree  of  accuracy  is  not  needed. 

Strips  of  thin  board  are  nailed  together,  as  shown 
in  Fig.  17,  to  form  a  triangle  with  equal  vertical  sides. 
The  mid-point  of  the  base  is  marked,  and  a  plumb 
line  is  let  fall  from  the  opposite  vertex.  When  the 
instrument  is  placed  so  that  the  line  crosses  the  mark, 


Fig.  17. 


the  bar  at  the  base  is  horizontal,  being  perpendicular  to 
the  plumb  line.  The  median  to  the  base  of  an  isosceles 
triangle  is  perpendicular  to  the  base.  From  the  lengths 
of  the  sides  of  the  triangle  it  may  be  computed — or  it 
may  be  found  by  trial — how  far  from  the  middle  of  the 
crossbar  a  mark  must  be  placed  so  that  when  the  plumb 
line  crosses  it  the  bar  shall  indicate  a  grade  of  1  in 
200,  1  in  100,  etc 


120 


"ROPE  STRETCHERS." 

If  a  rope  12  units  long  be  marked  off  into  three  seg- 
ments of  3,  4,  and  5  imits,  the  end  points  brought  to- 
gether, and  the  rope  stretched,  the  triangle  thus  formed 
is  right-angled  (Fig.  18).    This  was  used  by  the  build- 


Fig.  18. 

ers  of  the  pyramids.  The  Egyptian  word  for  sur- 
veyor means,  literally,  "rope  stretcher.''  Surveyors 
to  this  day  use  the  same  principle,  counting  off  some 
multiple  of  these  numbers  in  links  of  their  chain. 


121 


THE   THREE    FAMOUS    PROBLEMS   OF   AN- 
TIQUITY. 

1.  To  trisect  an  angle  or  arc. 

2.  To  ''duplicate  the  cube." 

3.  To  "square  the  circle." 

The  trisection  of  an  angle  is  an  ancient  problem ; 
"but  tradition  has  not  enshrined  its  origin  in  ro- 
mance."* The  squaring  of  the  circle  is  said  to  have 
been  first  attempted  by  Anaxagoras.  The  problem 
to  duplicate  the  cube  "was  known  in  ancient  times  as 
the  Delian  problem,  in  consequence  of  a  legend  that 
the  Delians  had  consulted  Plato  on  the  subject.  In 
one  form  of  the  story,  which  is  related  by  Philoporus, 
it  is  asserted  that  the  Athenians  in  430  B.  C,  when 
suffering  from  the  plague  of  eruptive  typhoid  fever, 
consulted  the  oracle  at  Delos  as  to  how  they  could 
stop  it.  Apollo  replied  that  they  must  double  the  size 
of  his  altar  which  was  in  the  form  of  a  cube.  To  the  un- 
learned suppliants  nothing  seemed  more  easy,  and  a  new 
altar  was  constructed  either  having  each  of  its  edges 
double  that  of  the  old  one  (from  which  it  followed 
that  the  volume  was  increased  eightfold)  or  by  pla- 
cing a  similar  cubic  altar  next  to  the  old  one.  Where- 
upon, according  to  the  legend,  the  indignant  god  made 
the  pestilence  worse  than  before,  and  informed  a 
fresh  deputation  that  it  was  useless  to  trifle  with  him, 
as  his  new  altar  must  be  a  cube  and  have  a  volume 

*  Ball,  Recreations,  p.  245. 


THE  THREE  PROBLEMS  OF  ANTIQUITY.  123 

exactly  double  that  of  his  old  one.  Suspecting  a  mys- 
tery the  Athenians  applied  to  Plato,  who  referred  them 
to  the  geometricians,  and  especially  to  Euclid,  who  had 
made  a  special  study  of  the  problem."*  It  is  a  hard- 
hearted historical  criticism  that  would  cast  a  doubt  on 
a  story  inherently  so  credible  as  this  on  account  of  so 
trifling  a  circumstance  as  that  Plato  was  not  born  till 
429  B.  C.  and  Euclid  much  later. 

Hippias  of  Elis  invented  the  quadratrix  for  the  tri- 
section  of  an  angle,  and  it  was  later  used  for  the 
quadrature  of  the  circle.  Other  Greeks  devised  other 
curves  to  effect  the  construction  required  in  (1)  and 
(2).  Eratosthenes  and  Nicomedes  invented  mechan- 
ical instruments  to  draw  such  curves.  But  none  of 
these  curves  can  be  constructed  with  ruler  and  com- 
pass alone.  And  this  was  the  limitation  imposed  on 
the  solution  of  the  problems. 

Antiquity  bequeathed  to  modern  times  all  three  prob- 
lems unsolved.  Modern  mathematics,  with  its  more 
efficient  methods,  has  proved  them  all  impossible  of 
construction  with  ruler  and  compass  alone — a  result 
which  the  shrewdest  investigator  in  antiquity  could 
have  only  conjectured — has  shown  new  ways  of  solv- 
ing them  if  the  limitation  of  ruler  and  compass  be  re- 
moved, and  has  devised  and  applied  methods  of  ap- 
proximation. It  has  dissolved  the  problems,  if  that 
term  may  be  permitted. 

It  was  not  until  1882  that  the  transcendental  nature 
of  the  number  -k  was  established  (by  Lindemann). 
The  final  results  in  all  three  of  the  problems,  with 
mathematical  demonstrations,  are  given  in  Klein's 
Famous  Problems  of  Elementary  Geometry.     A  more 

*  Ball,  Hist,  p.  43-4;  nearly  the  same  in  his  Recreations,  p. 
239-240. 


124    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

popular  and  elementary  discussion  is  Rupert's  Famous 
Geometrical  Theorems  and  Problems. 

It  should  be  noted  that  the  number  w9  which  the 
student  first  meets  as  the  ratio  of  the  circumference 
to  the  diameter  of  a  circle,  is  a  number  that  appears 
often  in  analysis  in  connections  remote  from  ele- 
mentary geometry ;  e.  g.,  in  formulas  in  the  calculus 
of  probability. 

The  value  of  ?r  was  computed  to  707  places  of  deci- 
mals by  William  Shanks.  His  result  (communicated 
in  1873)  with  a  discussion  of  the  formula  he  used 
(Machin's)  may  be  found  in  the  Proceedings  of  the 
Royal  Society  of  London,  Vol.  21.  No  other  problem 
of  the  sort  has  been  worked  out  to  such  a  degree  of 
accuracy — "an  accuracy  exceeding  the  ratio  of  micro- 
scopic to  telescopic  distances. "  An  illustration  calcu- 
lated to  give  some  conception  of  the  degree  of  accu- 
racy attained  may  be  found  in  Professor  Schubert's 
Mathematical  Essays  and  Recreations,  p.  140. 

Shanks  was  a  computer.  He  stands  in  contrast  to  the 
circle-squarers,  who  expect  to  find  a  "solution."  Most 
of  his  computation  serves,  apparently,  no  useful  pur- 
pose. But  it  should  be  a  deterrent  to  those  who — im- 
mune to  the  demonstration  of  Lindemann  and  others 
— still  hope  to  find  an  exact  ratio. 

The  quadrature  of  the  circle  has  been  the  most  fas- 
cinating of  mathematical  problems.  The  "army  of 
circle-squarers"  has  been  recruited  in  each  generation. 
"Their  efforts  remained  as  futile  as  though  they  had 
attempted  to  jump  into  a  rainbow"  (Cajori)  ;  yet  they 
were  undismayed.  In  some  minds,  the  proof  that  no 
solution  can  be  found  seems  only  to  have  lent  zest  to 
the  search. 


THE  THREE  PROBLEMS  OF  ANTIQUITY.  I25 

That  these  problems  are  of  perennial  interest,  is  at- 
tested by  the  fact  that  contributions  to  them  still  ap- 
pear. In  1905  a  little  book  was  published  in  Los 
Angeles  entitled  The  Secret  of  the  Circle  and  the 
Square,  in  which  also  the  division  of  "any  angle  into 
any  number  of  equal  angles''  is  considered.  The 
author,  J.  C.  Willmon,  gives  original  methods  of  ap- 
proximation. School  Science  and  Mathematics  for 
May  1906  contains  a  "solution"  of  the  trisection  prob- 
lem by  a  high-school  boy  in  Missouri,  printed,  appar- 
ently, to  show  that  the  problem  still  has  fascination 
for  the  youthful  mind.  In  a  later  number  of  that 
magazine  the  problem  is  discussed  by  another  from 
the  vantage  ground  of  higher  mathematics. 

While  the  three  problems  have  all  been  proved  to 
be  insolvable  under  the  condition  imposed,  still  the 
attempts  made  through  many  centuries  to  find  a  so- 
lution have  led  to  much  more  valuable  results,  not  only 
by  quickening  interest  in  mathematical  questions,  but 
especially  by  the  many  and  important  discoveries  that 
have  been  made  in  the  effort.  The  voyagers  were  un- 
able to  find  the  northwest  passage,  and  one  can  easily 
see  now  that  the  search  was  necessarily  futile  ;  but  in  the 
attempt  they  discovered  continents  whose  resources, 
whe.n  developed,  make  the  wealth  of  the  Indies  seem 
poor  indeed. 


THE   CIRCLE-SQUARER'S    PARADOX. 

Professor  De  Morgan,  in  his  Buaget  of  Paradoxes 
(London,  1872)  gave  circle-squarers  the  honor  of 
more  extended  individual  notice  and  more  complete 
refutation  than  is  often  accorded  them.  The  Budget 
first  appeared  in  instalments  in  the  Athen&um,  where 
it  attracted  the  correspondence  and  would-be  con- 
tributions of  all  the  circle-squarers,  and  the  like  ama- 
teurs, of  the  day.  His  facetious  treatment  of  them 
drew  forth  their  severest  criticisms,  which  in  turn 
gave  most  interesting  material  for  the  Budget.  He 
says  he  means  that  the  coming  New  Zealander  shall 
know  how  the  present  generation  regards  circle- 
squarers.  Theirs  is  one  of  the  most  amusing  of  the 
many  paradoxes  of  which  he  wrote.  The  book  is  out 
of  print,  and  so  rare  that  the  following  quotations  from 
it  may  be  acceptable: 

"Mere  pitch-and-toss  has  given  a  more  accurate 
approach  to  the  quadrature  of  the  circle  than  has  been 
reached  by  some  of  my  paradoxers .  .  .  The  method  is 
as  follows:  Suppose  a  planked  floor  of  the  usual  kind, 
with  thin  visible  seams  between  the  planks.  Let  there 
be  a  thin  straight  rod,  or  wire,  not  so  long  as  the 
breadth  of  the  plank.  This  rod,  being  tossed  up  at 
hazard,  will  either  fall  quite  clear  of  the  seams,  or 
will  lay  across  one  seam.  Now  Buffon,  and  after  him 
Laplace,  proved  the  following:  That  in  the  long  run 
the  fraction  of  the  whole  number  of  trials  in  which  a 

126 


THE   CIRCLE-SOUARER  S   PARADOX  127 

seam  is  intersected  will  be  the  fraction  which  twice 
the  length  of  the  rod  is  of  the  circumference  of  the 
circle  having  the  breadth  of  a  plank  for  its  diameter. 
In  1855  Mr.  Ambrose  Smith,  of  Aberdeen,  made 
3,204  trials  with  a  rod  three-fifths  of  the  distance 
between  the  planks:  there  were  1,213  clear  intersec- 
tions, and  11  contacts  on  which  it  was  difficult  to 
decide.  Divide  these  contacts  equally.  .  .  .this  gives 
7r  =  3-1553.  A  pupil  of  mine  made  600  trials  with  a 
rod  of  the  length  between  the  seams,  and  got  ir- 
3-137."    (P.  170-1.)* 

"The  celebrated  interminable  fraction  3-14159..., 
which  the  mathematician  calls  ?r,  is  the  ratio  of  the 
circumference  to  the  diameter.  But  it  is  thousands  of 
things  besides.  It  is  constantly  turning  up  in  mathe- 
matics :  and  if  arithmetic  and  algebra  had  been  studied 
without  geometry,  -n-  must  have  come  in  somehow, 
though  at  what  stage  or  under  what  name  must  have 
depended  upon  the  casualties  of  algebraical  invention. 
This  will  readily  be  seen  when  it  is  stated  that  -k  is 
nothing  but  four  times  the  series 

1      1_1      I      i_ 

ad  infinitum.  It  would  be  wonderful  if  so  simple  a 
series  had  but  one  kind  of  occurrence.  As  it  is,  our 
trigonometry  being  founded  on  the  circle,  -k  first  ap- 
pears as  the  ratio  stated.  If,  for  instance,  a  deep 
study  of  probable  fluctuation  from  the  average  had 
preceded  geometry,  ?r  might  have  emerged  as  a  number 
perfectly  indispensable  in  such  problems  as — What  is 

*  Ball,  in  his  Mathematical  Recreations  and  Essays  (p.  261, 
citing  the  Messenger  of  Mathematics,  Cambridge,  1873,  2 : 
1 13-4)  adds  that  "in  1864  Captain  Fox  made  1120  trials  with 
some  additional  precautions,  and  obtained  as  the  mean  value 

7T:=3.I4I9." 


128    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

the  chance  of  the  number  of  aces  lying  between  a 
million  +  x  and  a  million  -x,  when  six  million  of 
throws  are  made  with  a  die?"     (P.  171.) 

"More  than  thirty  years  ago  I  had  a  friend. .  .who 
was . .  .  thoroughly  up  in  all  that  relates  to  mortality, 
life  assurance,  etc.  One  day,  explaining  to  him  how  it 
should  be  ascertained  what  the  chance  is  of  the  sur- 
vivors of  a  large  number  of  persons  now  alive  lying 
between  given  limits  of  number  at  the  end  of  a  certain 
time,  I  came,  of  course,  upon  the  introduction  of  7r, 
which  I  could  only  describe  as  the  ratio  of  the  circum- 
ference of  a  circle  to  its  diameter.  'Oh,  my  dear 
friend !  that  must  be  a  delusion ;  what  can  the  circle 
have  to  do  with  the  numbers  alive  at  the  end  of  a  given 
time?' — 'I  cannot  demonstrate  it  to  you;  but  it  is 
demonstrated.'  "     (P.  172.) 

"The  feeling  which  tempts  persons  to  this  problem 
[exact  quadrature]  is  that  which,  in  romance,  made  it 
impossible  for  a  knight  to  pass  a  castle  which  belonged 
to  a  giant  or  an  enchanter.  I  once  gave  a  lecture  on 
the  subject:  a  gentleman  who  was  introduced  to  it  by 
what  I  said  remarked,  loud  enough  to  be  heard  all 
around,  'Only  prove  to  me  that  it  is  impossible,  and  I 
will  set  about  it  this  very  evening.' 

"This  rinderpest  of  geometry  cannot  be  cured,  when 
once  it  has  seated  itself  in  the  system:  all  that  can  be 
done  is  to  apply  what  the  learned  call  prophylactics  to 
those  who  are  yet  sound."     (P.  390.) 

"The  finding  of  two  mean  proportionals  is  the  pre- 
liminary to  the  famous  old  problem  of  the  duplica- 
tion of  the  cube,  proposed  by  Apollo  (not  Apollonius) 
himself.  DTsraeli  speaks  of  the  'six  follies  of  science/ 
— the  quadrature,  the  duplication,  the  perpetual  mo- 
tion, the   philosopher's   stone,   magic,   and   astrology. 


THE   CIRCLE-SQUARER'S   PARADOX  129 

He  might  as  well  have  added  the  trisection,  to  make  the 
mystic  number  seven :  but  had  he  done  so,  he  would 
still  have  been  very  lenient ;  only  seven  follies  in  all 
science,  from  mathematics  to  chemistry !  Science  might 
have  said  to  such  a  judge — as  convicts  used  to  say 
who  got  seven  years,  expecting  it  for  life,  'Thank 
you,  my  Lord,  and  may  you  sit  there  till  they  are  over,' 
— mav  the  Curiosities  of  Literature  outlive  the  Follies 
of  Science!"     (P.  71.) 


THE  INSTRUMENTS  THAT  ARE  POSTU- 
LATED. 

The  use  of  two  instruments  is  allowed  in  theoretic 
elementary  geometry,  the  ruler  and  the  compass — a 
limitation  said  to  be  due  to  Plato. 

It  is  understood  that  the  compass  is  to  be  of  un- 
limited opening.  For  if  the  compass  would  not  open 
as  far  as  we  please,  it  could  not  be  used  to  effect  the 
construction  demanded  in  Euclid's  third  postulate,  the 
drawing  of  a  circle  with  any  center  and  any  radius. 
Similarly,  it  is  understood  that  the  ruler  is  of  unlimited 
length  for  the  use  of  the  second  postulate. 

Also  that  the  ruler  is  ungraduated.  If  there  were 
even  two  marks  on  the  straight-edge  and  we  were 
allowed  to  use  these  and  move  the  ruler  so  as  to  tit  a 
figure,  the  problem  to  trisect  an  angle  (impossible 
to  elementary  geometry)  could  be  readily  solved,  as 
follows : 

Let  ABC  be  the  angle,  and  P,  Q  the  two  points 
on  the  straight-edge.     (Fig.   19.) 

On  one  arm  of  angle  B  lav  off  BA  =  PQ.  Bisect 
BA,  at  M.. 

Draw  MK||BC,  and  ML  1  BC. 

Adjust  the  straight-edge  to  fit  the  figure  so  that 
P  lies  on  MK,  Q  on  ML,  and  at  the  same  time  the 
straight-edge  passes  through  B.  Then  BP  trisects 
the  angle. 


130 


1 X STRU MEN TS  POSTULATED. 


131 


Proof.    ZPBC  =  its  alternate  ZBPM. 
Mark  N  the  mid-point  of  PQ,  and  draw  NM.    Then 
X,  the  mid-point  of  the  hypotenuse  of  the  rt.  A  POM, 
is  equidistant  from  the  vertexes  of  the  triangle. 
.-.ZBPM  =  ZPMN 
Exterior  ZBNM  =  ZBPM  4-  ZPMN 
=  2ZBPM 


Q 


Fig.  19. 


•.•NM  =  JPQ  =  BM 
:-.ZMBN  =  ZBNM 
ZPBC  -  ZBPM  =  iZBNM  =  JZABN  =  iZABG 


132    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

A.  B.  Kempe,*  from  whom  this  form  of  the  well- 
known  solution  is  adapted,  raises  the  question  whether 
Euclid  does  not  use  a  graduated  ruler  and  the  fitting 
process  when,  in  book  1,  proposition  4,  he  fits  side 
AB  of  triangle  ABr  to  side  AE  of  triangle  AEZ — 
the  first  proof  by  superposition,  with  which  every  high- 
school  pupil  is  familiar.  It  may  be  replied  that  Euclid 
does  not  determine  a  point  (as  P  is  found  in  the 
angle  above)  by  fitting  and  measuring.  He  super- 
poses only  in  his  reasoning,  in  his  proof. 

Our  straight-edge  must  be  ungraduated,  or  it  grants 
us  too  much ;  it  must  be  unlimited  or  it  grants  us  too 
little. 

*  How  to  Draw  a  Straight  Line,  note  (2). 


THE  TRIANGLE  AND  ITS  CIRCLES. 

The  following  statement  of  notation  and  familiar 
definitions  may  be  permitted : 

0,  orthocenter  of  the  triangle  ABD,  the  point  of 
concurrence  of  the  three  altitudes  of  the  triangle. 

G,  center  of  gravity,  center  of  mass,  or  centroid,  of 
the  triangle,  the  point  of  concurrence  of  the  three 
medians. 

C,  circnmcenter  of  the  triangle,  center  of  the  cir- 
cumscribed circle,  point  of  concurrence  of  the  per- 
pendicular bisectors  of  the  sides  of  the  triangle.     . 

1,  in-center  of  the  triangle,  center  of  the  inscribed 
circle,  point  of  concurrence  of  the  bisectors  of  the 
three  interior  angles  of  the  triangle. 

E,  E,  E,  ex-centers,  centers  of  the  escribed  circles, 
each  E  the  point  of  concurrence  of  the  bisectors  of 
two  exterior  angles  of  the  triangle  and  one  interior 
angle. 

An  obtuse  angled  triangle  is  used  in  the  figure  so 
tnat  the  centers  may  be  farther  apart  and  the  figure 
less  crowded. 

Collinearity  of  centers.  O,  G,  and  C  are  collinear, 
and  OG  =  twice  GC. 

Corollary:  The  distance  from  O  to  a  vertex  of  the 
triangle  is  twice  the  distance  from  C  to  the  side  oppo- 
site that  vertex.* 

*  Or  this  corollary  may  easily  be  proved  independently  and 
the  proposition  that  O,  G  and  C  are  in  a  straight  line  of  which 

133 


134    A  SCRAP-BOOK  OF  ELEMENTARY   MATHEMATICS. 

The  nine-point  circle.     Let  L,   M,   N  be  the  mid- 
points of  the  sides ;  A',  B',  D',  the  projections  of  the 


Fig.  20. 


vertexes  on  the  opposite  sides;  H,  J,  K,  the  mid-points 
of  OA,  OB,  OD,  respectively.    Then  these  nine  points 

G  is  a  trisection  point  be  derived  from  it,  as  the  writer  once  did 
when  unacquainted  with  me  results  that  had  been  achieved 
in  this  field. 


THE  TRIANGLE  AND  ITS  CIRCLES.  I35 

are  concyclic ;  and  the  circle  through  them  is  called 
the  nine-pcint  circle  of  the  triangle   (Fig.  20). 

The  center  of  the  nine-point  circle  is  the  mid-point 
of  OC,  and  its  radius  is  half  the  radius  of  the  circum- 
scribed circle. 

The  discovery  of  the  nine-point  circle  has  been  er- 
roneously attributed  to  Euler.  Several  investigators 
discovered  it  independently  in  the  early  part  of  the 
nineteenth  century.  The  name  nine- point  circle  is  said 
to  be  due  to  Terquem  (1842)  editor  of  Nouvelles  An- 
nates. Karl  Wilhelm  Feuerbach  proved,  in  a  pamphlet 
of  1822,  what  is  now  known  as  "Feuerbach's  theorem": 
The  nine-point  circle  of  a  triangle  is  tangent  to  the 
inscribed  circle  and  each  of  the  escribed  circles  of  the 
triangle. 

So  many  beautiful  theorems  about  the  triangle  have 
been  proved  that  Crelle — himself  one  of  the  foremost 
investigators  of  it — exclaimed :  "It  is  indeed  wonderful 
that  so  simple  a  figure  as  the  triangle  is  so  inexhaust- 
ible in  properties.  How  many  as  yet  unknown  prop- 
erties of  other  figures  may  there  not  be!" 

The  reader  is  referred  to  Cajori's  History  of  Ele- 
mentary Mathematics  and  the  treatises  on  this  subject 
mentioned  in  his  note,  p.  259,  and  to  the  delightful 
monograph,  Some  Notezvorthy  Properties  of  the  Tri- 
angle and  Its  Circles,  by  W.  H.  Bruce,  president  of 
the  North  Texas  State  Normal  School,  Denton.  Many 
of  Dr.  Bruce's  proofs  and  some  of  his  theorems  are 
original. 


LINKAGES   AND    STRAIGHT-LINE   MOTION. 

Under  the  title  How  to  Draw  a  Straight  Line,  A. 
B.  Kempe  wrote  a  little  book  which  is  full  of  theoretic 
interest  to  the  geometer,  as  it  touches  one  of  the  foun- 
dation postulates  of  the  science. 

We  occasionally  run  a  pencil  around  a  coin  to  draw 
a  circumference,  thus  using  one  circle  to  produce  an- 
other. But  this  is  only  a  makeshift :  we  have  an  in- 
strument, not  itself  circular,  with  which  to  draw  a 
circle — the  compass.  Now,  when  we  come  to  draw 
a  straight  line  we  say  that  that  postulate  grants  us  the 


Fig.  21. 

use  of  a  ruler.  But  this  is  demanding  a  straight  edge 
for  drawing  a  straight  line — given  a  straight  line  to 
copy.  Is  it  possible  to  construct  an  instrument,  not 
itself  straight,  which  shall  draw  a  straight  line?  Such 
an  instrument  was  first  invented  by  Peaucellier,  a 
French  army  officer  in  the  engineer  corps.  It  is  a 
"linkage."  Since  that  time  (1864)  other  linkages 
have  been  invented  to  effect  rectilinear  motion,  some 
of  them  simpler  than  Peaucellier's.  But  as  his  is 
earliest,  it  may  be  taken  as  the  type. 

Preliminary  to  its  construction,  however,  let  us  con- 

136 


LINKAGES  AND  STRAIGHT-LINE  MOTION. 


137 


sider  a  single  link  (Fig.  21)  pivoted  at  one  end  and 
carrying  a  pencil  at  the  other.  The  pencil  describes 
a  circumference.  If  two  links  (Fig.  22)  be  hinged 
at  H,  and  point  F  fastened  to  the  plane,  point  P  is 


Fig.  25. 


Fig.  26. 


free  to  move  in  any  direction  ;  its  path  is  indetermin- 
ate. The  number  of  links  must  be  odd  to  give  de- 
terminate   motion.      If    a    system    of   three    links    be 


I38    A  SCRAP-BOOK  OF  ELEMENTARY   MATHEMATICS. 

fastened  at  both  ends,  a  point  in  the  middle  link  de- 
scribes a  definite  curve — say  a  loop.  Five  links  can 
give  the  requisite  straight-line  motion ;  but  Peaucel- 
lier's  was  a  seven-link  apparatus. 

Such  a  linkage  can  be  made  by  any  teacher.  The 
writer  once  made  a  small  one  of  links  cut  out  of  card- 
board and  fastened  together  by  shoemaker's  eyelets ; 
also  a  larger  one  (about  30  times  the  size  of  Fig.  23) 
of  thin  boards  joined  with  bolts.  F  and  O  (Fig.  23) 
were  made  to  fasten  in  mouldings  above  the  black- 
board, and  P  carried  a  piece  of  crayon.  This  proved 
very  interesting  to  a  geometry  class  for  a  lecture.  It 
is  needless  to  say  that  no  one  would  think  of  any  such 
appliance  for  daily  class-room  use.  The  ruler  is  the 
practical  instrument. 

Fig.  24  is  a  diagram  of  the  apparatus  shown  in 
Fig.  23.  FA  =  FB.  In  all  positions  APBC  is  a  rhom- 
bus. F  and  O  are  fastened  at  points  whose  distance 
apart  is  equal  to  OC.  Then  C  moves  in  an  arc  of  a 
circle  whose  center  is  O ;  A  and  B  move  in  an  arc 
with  center  at  F.  It  is  to  be  shown  that  P  moves  in  a 
straight  line. 

Draw  PP'  1  FO  produced.*     FCC  being  inscribed 
in  a  semicircle,  is  a  right  angle.    Hence  As  FP'P  and 
FCC,  having  ZF  in  common,  are  similar,  and 
pp..  fP  =  FC  :  FC 
FP  •  FC  =FP'  •  FC  (1) 

F,  C  and  P,  being  each  equidistant  from  A  and  B, 
lie  in  the  same  straight  line ;  and  the  diagonals  of  the 
rhombus  APBC  are  perpendicular  bisectors  of  each 
other.     Hence 

*  Imagine  these  lines  drawn,  if  one  objects  to  drawing  a 
straight  line  as  one  step  in  the  process  of  showing  that  a 
straight  line  can  be  drawn ! 


LINKAGES  AND  STRAIGHT-LINE   MOTION.  I39 

FB2  =  FM2  +  MB2 
PB2  =  MP2  +  MB2 

.-.  FB2  -  PB2  =  FM2  -  MP2 

=  (FM  +  MP)(FM-MP) 
=  FP-FC  (2) 

From  (1)  and  (2),  FP'  •  FC'  =  FB2  -  PB2. 

But  as  the  linkage  moves,  FC\  FB  and  PB  all  re- 
main constant;  therefore  FP'  is  constant.  That  is, 
P',  the  projection  of  P  on  FO,  is  always  the  same 
point;  or  in  other  words,  P  moves  in  a  straight  line 
(perpendicular  to  FO). 

If  the  distance  between  the  two  fixed  points,  F  and 
O,  be  made  less  than  the  length  of  the  link  OC,  P 
moves  in  an  arc  of  a  circle  with  concave  toward  O 
(Fig.  25).  As  OC-OF  approaches  zero  as  a  limit, 
the  radius  of  the  arc  traced  by  P  increases  without 
limit. 

Then  as  would  be  expected,  if  OF  be  made  greater 
than  OC,  P  traces  an  arc  that  is  convex  toward  O 
(Fig.  26).  The  smaller  OF-OC,  the  longer  the 
radius  of  the  arc  traced  by  P.  It  is  curious  that  so 
small  an  instrument  may  be  used  to  describe  an  arc 
of  a  circle  with  enormous  radius  and  with  center  on 
the  opposite  side  of  the  arc  from  the  instrument. 

The  straight  line — the  "simplest  curve"  of  mathe- 
maticians— lies  between  these  two  arcs,  and  is  the 
limiting  form  of  each. 

Linkages  possess  many  interesting  properties.  The 
subject  was  first  presented  to  English-speaking  stu- 
dents by  the  late  Professor  Sylvester.  Mr.  Kempe 
showed  "that  a  link-motion  can  be  found  to  describe 
any  given  algebraic  curve.,, 


THE  FOUR-COLORS  THEOREM. 

This  theorem,  known  also  as  the  map  makers' 
proposition,  has  become  celebrated.  It  is,  that  four 
colors  are  sufficient  for  any  map,  no  two  districts  hav- 
ing a  common  boundary  line  to  be  colored  the  same ; 
and  this  no  matter  how  numerous  the  districts,  how 
irregular  their  boundaries  or  how  complicated  their 
arrangement. 

That  four  colors  may  be  necessary  can  be  seen  from 
Fig.  27.  A  few  trials  will  con- 
vince most  persons  that  it  is  prob- 
ably impossible  to  draw  a  map  re- 
quiring more  than  four.  To  give 
a  mathematical  proof  of  it,  is  quite 
another  matter. 

The  proposition  is  said  to  have 
been  long  known  to  map  makers.  It  was  mentioned 
as  a  mathematical  proposition  by  A.  F.  Mobius,  in 
1840,  and  later  popularized  by  De  Morgan.  All  that 
is  needed  to  give  a  proposition  celebrity  is  to  proclaim 
it  one  of  the  unsolved  problems  of  the  science.  Cay- 
ley's  remark,  in  1878,  that  this  one  had  remained 
unproved  was  followed  by  at  least  two  published  dem- 
onstrations within  two  or  three  years.  But  each  had  a 
flaw.  The  chance  is  still  open  for  some  one  to  invent 
a  new  method  of  attack. 

If  the  proposition   were  not  true,  it  could  be  dis- 
proved by  a  single  special  case,  by  producing  a  "map" 

140 


THE  FOUR   COLORS  THEOREM. 


HI 


with  five  districts  of  which  each  bounds  every  other. 
Many  have  tried  to  do  this. 

It  has  been  shown  that  there  are  surfaces  on  which 
the  proposition  would  not  hold  true.  The  theorem 
refers  to  a  plane  or  the  surface  of  a  globe. 

For  historical  presentation  and  bibliographic  notes, 
see  Ball's  Recreations,  pp.  51-3;  or  for  a  more  ex- 
tended discussion,  Lucas,  IV,  168  et  seq. 


PARALLELOGRAM  OF  FORCES. 

One  of  the  best-known  principles  of  physics  is,  that 
if  a  ball,  B,  is  struck  a  blow  which  if  acting  alone 
would  drive  the  ball  to  A,  and  a  blow  which  alone 
would  drive  it  to  C,  and  both  blows  are  delivered  at 
once,  the  ball  takes  the  direction  BD,  the  diagonal  of 

the   parallelogram   of    BA 
A  D     and   BC,  and  the   force  is 

just  sufficient  to  drive  the 
ball  to  D.     BD  is  the  re- 
•g  q  sultant  of  the  two  forces. 

Pig  2g  If  a  third   force,   repre- 

sented by  some  line  BE, 
operates  simultaneously  with  those  represented  by  BA 
and  BC,  then  the  diagonal  of  the  parallelogram  of  BD 
and  BE  is  the  resultant  of  the  three  forces.  And 
so  on. 

Hence  the  resultant  of  forces  is  always  less  than 
the  sum  of  the  forces  unless  the  forces  act  in  the  same 
direction.  The  more  nearly  their  lines  of  action  ap- 
proach each  other,  the  more  nearly  does  their  resultant 
approach  their  sum. 

One  is  tempted  to  draw  the  moral,  that  social  forces 
have  a  resultant  and  obey  an  analogous  law,  the  result 
of  all  the  educational  or  other  social  energy  expended 
on  a  child,  or  in  a  community,  being  less  than  the 
sum,  unless  all  forces  act  in  the  same  line. 


T42 


A  QUESTION   OF   FOURTH   DIMENSION   BY 

ANALOGY. 

After  class  one  day  a  normal-school  pupil  asked  the 
writer  the  following  question,  and  received  the  follow- 
ing reply : 

Q.  If  the  path  of  a  moving  point  (no  dimension) 
is  a  line  (one  dimension),  and  the  path  of  a  moving 
line  is  a  surface  (two  dimensions),  and  the  path  of 
a  moving  surface  is  a  solid  (three  dimensions),  why 
isn't  the  path  of  a  moving  solid  a  four-dimensional 
magnitude  ? 

A:  If  your  hypotheses  were  correct,  your  conclusion 
should  follow  by  analogy.  The  path  of  a  moving  point 
is,  indeed,  always  a  line.  The  path  of  a  moving  line 
is  a  surface  except  when  the  line  moves  in  its  own 
dimension,  "slides  in  its  trace. "  The  path  of  a  moving 
surface  is  a  solid  only  when  the  motion  is  in  a  third 
dimension.  The  generation  of  a  four  -  dimensional 
magnitude  by  the  motion  of  a  solid  presupposes  that 
the  solid  is  to  be  moved  in  a  fourth  dimension. 


SYMMETRY  ILLUSTRATED  BY  PAPER 
FOLDING. 

The  following  simple  device  has  been  found  by  the 
writer  'to  give  pupils  an  idea  of  symmetry  with  a 
certainty  and  directness  which  no  verbal  explanation 
unaided  can  approach.  Require  each  pupil  to  take 
a  piece  of  calendered  or  sized  paper,  fold  and  crease 
it  once,  straighten  it  out  again,  draw  rapidly  with  ink 
any  figure  on  one  half  of  the  paper,  and  fold  together 
while  the  ink  is  still  damp.  The  original  drawing  and 
the  trace  on  the  other  half  of  the  paper  are  symmetric 
with  respect  to  the  crease  as  an  axis.  Again:  Fold 
a  paper  in  two  perpendicular  creases.  In  one  quadrant 
draw  a  figure  whose  two  end  points  lie  one  in  each 
crease.  Quickly  fold  so  as  to  make  a  trace  in  each 
of  the  other  quadrants.  A  closed  figure  is  formed 
which  is  symmetric  with  respect  to  the  intersection 
of  the  creases  as  center. 

T.  Sundara  Row,  in  his  Geometric  Exercises  in 
Paper  Folding  (edited  and  revised  by  Beman  and 
Smith),*  has  shown  how  to  make  many  of, the  con- 
structions of  plane  geometry  by  paper  folding,  in- 
cluding beautiful  illustrations  of  some  of  the  regular 
polygons  and  the  locating  of  points  on  some  of  the 
higher  plane  curves. 

*  Chicago,  The  Open  Court  Publishing  Co. 


M4 


SYMMETRY.  I45 

Illustrations  of  symmetry  by  the  use  of  the  mirror 
are  well  brought  out  in  a  brief  article  recently  pub- 
lished in  American  Education  * 

*  Number   for   March    1907,   p.   464-5,   article   "Symmetrical 
Plane  Figures,"  by  Henry  J.  Lathrop. 


APPARATUS  TO  ILLUSTRATE  LINE  VALUES 
OF  TRIGONOMETRIC  FUNCTIONS. 

A  piece  of  apparatus  to  illustrate  trigonometric 
lines  representing  the  trigonometric  ratios  may  be 
constructed  somewhat  as  follows   (Fig.  29)  : 

To  the  center  O  of  a  disc  is  attached  a  rod  OR, 
which  may  be  revolved.     A  tangent  rod  is  screwed 


R 


Fig.  29. 


to  the  disc  at  A.  Along  this  a  little  block  bearing  the 
letter  T  is  made  to  slide  easily.  The  block  is  also 
connected  to  the  rod  OR,  so  that  T  marks  the  inter- 
section of  the  two  lines.  Similarly  a  block  R  is  moved 
along  the  tangent  rod  BR.  At  P,  a  unit's  distance 
from  O  on  the  rod  OR,  another  rod  (PM)  is  pivoted. 

1 46 


LINE  VALUES  OF  TRIGONOMETRIC  FUNCTIONS      I47 

A  weight  at  the  lower  end  keeps  the  rod  in  a  vertical 
position.  It  passes  through  a  block  which  is  made  to 
slide  freely  along  OA  and  which  bears  the  letter  M. 

As  the  rod  OR  is  revolved  in  the  positive  direction, 
increasing  the  angle  O,  MP  represents  the  increasing 
sine,  OM  the  decreasing  cosine,  AT  the  increasing 
tangent,  BR  the  decreasing  cotangent,  OT  the  increas- 
ing secant,  OR  the  decreasing  cosecant. 


"SINE/' 

Students  in  trigonometry  sometimes  say:  "From 
the  line  value,  or  geometric  representation,  of  the 
trigonometric  ratios  it  is  easy  to  see  why  the  tangent 
and  secant  were  so  named.  And  the  co-functions  are 
the  functions  of  the  complementary  angles.  But  what 
is  the  origin  of  the  name  sine?"  It  is  a  good  ques- 
tion. The  following  answer  is  that  of  Cantor,  Fink, 
and  Cajori ;  but  Cantor  deems  it  doubtful. 

The  Greeks  used  the  entire  chord  of  double  the  arc. 
The  Hindus,  though  employing  half  the  chord  of 
double  the  arc  (what  we  call  sine  in  a  unit  circle); 
used  for  it  their  former  name  for  the  entire  chord, 
jiva,  which  meant  literally  "bow-string,"  a  natural 
designation  for  chord.  Their  work  came  to  us  through 
the  Arabs,  who  transliterated  the  Sanskrit  jiva  into 
Arabic  dschiba.  Arabic  being  usually  written  in  "un- 
pointed text"  (without  vowels)  like  a  modern  sten- 
ographer's notes,  dschiba  having  no  meaning  in  Ar- 
abic, and  the  Arabic  word  dschaib  having  the  same 
consonants,  it  was  easy  for  the  latter  to  take  the  place 
of  the  former.  But  dschaib  means  "bosom."  Al  Bat- 
tani,  the  foremost  astronomer  of  the  ninth  century, 
wrote  a  book  on  the  motion  of  the  heavenly  bodies. 
In  the  twelfth  century  this  was  translated  into  Latin 
by  Plato  Tiburtinus,  who  rendered  the  Arabic  word 
by  the  Latin  sinus  (bosom).  And  sinus,  Anglicized,  is 
"sine." 

i48 


GROWTH  OF  THE  PHILOSOPHY  OF  THE 
CALCULUS. 

The  latter  half  of  the  seventeenth  century  produced 
that  powerful  instrument  of  mathematical  research, 
the  differential  calculus. *  The  master  minds  that  in- 
vented it,  Newton  and  Leibnitz,  failed  to  clear  the 
subject  of  philosophical  difficulties. 

Newton's  reasoning  is  based  on  this  initial  theorem 
in  the  Principia:  "Quantities,  and  the  ratios  of  quan- 
tities, that  during  any  finite  time  constantly  approach 
each  other,  and  before  the  end  of  that  time  approach 
nearer  than  any  given  difference,  are  ultimately 
equal."  It  is  not  surprising  that  neither  this  state- 
ment nor  its  demonstration  gave  universal  satisfac- 
tion.    The  "zeros"  whose  ratio  was  considered  in  the 

*  Newton  and  Leibnitz  invented  it  in  the  sense  that  they 
brought  it  to  comparative  perfection  as  an  instrument  of  re- 
search. Like  most  epoch-making  discoveries  it  had  been  fore- 
shadowed. Cavalieri,  Kepler,  Fermat  and  many  others  had 
been  working  toward  it  One  must  go  a  long  way  back  into 
the  history  of  mathematics  to  find  a  time  when  there  was  no 
suggestion  of  it.  As  this  note  is  penned  the  newspapers  bring 
a  report  that  Mr.  Hiberg,  a  Danish  scientist,  says  he  has  re- 
cently discovered  in  a  palimpsest  in  Constantinople,  a  hitherto 
unknown  work  on  mathematics  by  Archimedes.  "The  manu- 
script, which  is  entitled  'On  Method/  is  dedicated  to  Eratos- 
thenes, and  relates  to  the  applying  of  mechanics  to  the  solu- 
tion of  certain  problems  in  geometry.  There  is  in  this  ancient 
Greek  manuscript  a  method  that  bears  a  strong  resemblance  to 
the  integral  calculus  of  modern  days,  and  is  capable  of  being 
used  for  the  solution  of  problems  reserved  for  the  genius  of 
Leibnitz  and  Newton  eighteen  centuries  later."  (N.  Y.  Trib- 
une. ) 

149 


I50    A  SCRAP-BOOK  OF  ELEMENTARY   MATHEMATICS. 

method  of  fluxions  were  characterized  by  the  astute 
Bishop  Berkeley  as   "ghosts  of  departed  quantities." 

Leibnitz  based  his  calculus  on  the  principle  that 
one  may  substitute  for  any  magnitude  another  which 
differs  from  it  only  by  a  quantity  infinitely  small. 
This  is  assumed  as  "a  sort  of  axiom."  Pressed  for  an 
explanation,  he  said  that,  in  comparison  with  finite 
quantities,  he  treated  infinitely  small  quantities  as 
incomp  arables,  negligible  "like  grains  of  sand  in  com- 
parison with  the  sea."  This,  if  consistently  held, 
should  have  made  the  calculus  a  mere  method  of  ap- 
proximation. 

According  to  the  explanations  of  both,  strictly  ap- 
plied, the  calculus  should  have  produced  results  that 
were  close  approximations.  But  instead,  its  results 
were  absolutely  accurate.  Berkeley  first,  and  after- 
ward L.  N.  M.  Carnot,  pointed  out  that  this  was  due 
to  compensation  of  errors.  This  phase  of  the  subject 
is  perhaps  nowhere  treated  in  a  more  piquant  style 
than  in  Bledsoe's  Philosophy  of  Mathematics. 

The  method  of  limits  permits  a  rigor  of  demonstra- 
tion not  possible  to  the  pure  infinitesimalists.  Logic- 
ally the  methods  of  the  latter  are  to  be  regarded  as 
abridgments.  As  treated  by  the  best  writers  the  cal- 
culus is  to-day  on  a  sound  philosophical  basis.  It  is 
admirable  for  its  logic  as  well  as  for  its  marvelous 
efficiency. 

But  many  writers  are  so  dominated  by  the  thinking 
of  the  past  that  they  still  use  the  symbol  0  to  mean 
sometimes  "an  infinitely  small  quantity"  and  some- 
times absolute  zero.  Clearer  thinking  impels  to  the 
use  of  t  (iota)  or  i  or  some  other  symbol  to  mean  an 
infinitesimal,  denoting  by  0  only  zero. 

This  distinction  implies  that  between  their  recipro- 


PHILOSOPHY  OF  THE  CALCULUS.  I5I 

cals.  The  symbol  »  ,  first  used  for  an  infinite  by 
Wallis  in  the  seventeenth  century,  has  long  been  used 
both  for  a  variable  increasing  without  limit  and  for 
absolute  infinity.  The  revised  edition  of  Taylor's 
Calculus  (Ginn  1898)  introduced  a  new  symbol  op, 
a  contraction  of  a/0,  for  absolute  infinity,  using  » 
only  for  an  infinite  (the  reciprocal  of  an  infinitesimal). 
It  is  to  be  hoped  that  this  usage  will  become  universal. 
In  the  book  just  referred  to  is  perhaps  the  clearest 
and  most  concise  statement  to  be  found  anywhere  of 
the  inverse  problems  of  the  differential  calculus  and 
the  integral  calculus,  as  well  as  of  the  three  methods 
used  in  the  calculus. 


SOME  ILLUSTRATIONS  OF  LIMITS. 

Physical  illustrations  of  variables  are  numerous.  But 
to  find  a  similar  case  of  a  constant,  is  not  easy.  The 
long  history  of  the  determination  of  standards  (yard, 
meter  etc.)  is  the  history  of  a  search  for  physical  con- 
stants. Constants  are  the  result  of  abstraction  or  are 
limited  by  definition.  Non-physical  constants  are  nu- 
merous, and  enter  into  most  problems. 

If  one  person  is  just  a  year  older  than  another,  the 

ratio  of  the  age  of  the  younger  to  that  of  the  older,  at 

.       -  .  -u  -  .01234  49   50 

successive  birthdays,  1S  ---   -    -...__... 

In  general :  the  ratio  of  the  ages  of  any  two  persons 
is  a  variable  approaching  unity  as  limit.  The  sum  of 
of  their  ages  .is  a  variable  increasing  without  limit. 
The  difference  between  their  ages  is  a  constant. 

1 1 1 1 

A  P  P  P  B 

Fig.  30. 

When  pupils  have  the  idea  of  the  time-honored 
point  P  which  moves  half  way  from  A  to  B  the  first 
second,  half  the  remaining  distance  the  next  second, 
etc.,  but  have  trouble  with  the  product  of  a  constant 
and  a  variable,  they  have  sometimes  been  helped  by  the 
following  "optical  illustration" :  Imagine  yourself  look- 
ing at  Fig.  30  through  a  glass  that  makes  everything 
look  twice  as  large  as  it  appears  to  the  naked  eye. 

152 


SOME   ILLUSTRATIONS  OF   LIMITS. 


153 


AP  still  seems  to  approach  AB  as  limit ;  that  is,  twice 
the  "real"  AP  is  approaching  twice  the  "real"  AB  as 
limit.  Now  suppose  your  glass  magnifies  3  times, 
n  times.  AP  still  approaches  AB  magnified  the  same 
number  of  times.  That  is,  if  AP  =  AB,  then  any 
constant  x  AP  =  that  constant  x  AB. 

Reverse  the  glass,  making  AP  look  one-nth  part  as 
large  as  at  first.  It  approaches  one-nth  of  the  "real" 
AB.  Putting  this  in  symbols,  with  x  representing 
the  variable,  and  c  the  constant, 

■U§JU 

Or  in  words:  The  limit  of  the  ratio  of  a  variable  to  a 
constant  is  the  ratio  of 
the  limit  of  the  variable 
to  the  constant. 

Let  x  represent  the 
broken  line  from  A  to  C 
(Fig.  31),  composed 
first  of  4  parts,  then  of 
8,  then  of  16  (the  last 
division  shown  in  the 
figure)  then  of  32,  etc. 
The  polygon  bounded 
by  x,  AB  and  BC  = 
A  ABC  What  of  the 
length  of  x?  Most  per- 
sons to  whom  this  old  figure  is  new  answer  off-hand, 
"x  =  AC."  But  a  minute's  reflection  shows  that  x 
is  constant  and  =  AB  +  BC. 


Fig.  31. 


LAW  OF  COMMUTATION. 

This  law,  emphasized  for  arithmetic  in  McLellan 
and  Dewey's  Psychology  of  Number,  and  explicitly 
employed  in  all  algebras  that  give  attention  to  the 
logical  side  of  the  subject,  is  one  whose  importance 
is  often  overlooked.  So  long  as  it  is  used  implicitly 
and  regarded  as  of  universal  application,  its  import 
is  neglected.  An  antidote :  to  remember  that  there  are 
regions  in  which  this  law  does  not  apply.     E.  g. : 

In  the  "geometric  multiplication"  of  rectangular  vec- 
tors used  in  quaternions,  the  commutative  property 
of  factors  does  not  hold,  but  a  change  in  the  cyclic 
order  of  factors  reverses  the  sign  of  the  product. 

Even  in  elementary  algebra  or  arithmetic,  the  com- 
mutative principle  is  not  valid  in  the  operation  of  in- 
volution. Professor  Schubert,  in  his  Mathematical 
Essays  and  Recreations,  has  called  attention  to  the 
fact  that  this  limitation — the  impossibility  of  inter- 
changing base  and  exponent — renders  useless  any  high 
operation  of  continued  involution. 


*54 


EQUATIONS   OF   U.    S.    STANDARDS    OF 
LENGTH  AND  MASS. 

By  order  approved  by  the  secretary  of  the  treasury 
April  5,  1893,  the  international  prototype  meter  and 
kilogram  are  regarded  as  fundamental  standards,  the 
yard,  pound  etc.  being  defined  in  terms  of  them. 

All  of  the  nations  taking  part  in  the  convention 
have  very  accurate  copies  of  the  international  stan- 
dards. The  degree  of  accuracy  of  the  comparisons 
may  be  seen  from  the  equations  expressing  the  rela- 
tion of  meter  no. 27  and  kilogram  no. 20,  of  the 
United  States,  to  the  international  prototypes.  T  rep- 
resents the  number  of  degrees  of  the  centigrade  scale 
of  the  hydrogen  thermometer.  The  last  term  in  each 
equation  shows  the  range  of  error. 
M  no. 27=  lm-  1.6/*  +  8.657//T  +  0.00100/*T2±0.2/* 
K  no .  20  =  1kg  -  0.039  mg  ±  0.002  mg 

(U.  S.  coast  and  geodetic  survey.) 


THE  MATHEMATICAL  TREATMENT  OF 
STATISTICS. 

This  is  one  of  the  most  important  and  interesting 
applications  of  mathematics  to  the  needs  of  modern 
civilization.  Just  as  data  gathered  by  an  incompetent 
observer  are  worthless — or  by  a  biased  observer,  unless 
the  bias  can  be  measured  and  eliminated  from  the  re- 
sult— so  also  conclusions  obtained  from  even  the  best 
data  by  one  unacquainted  with  the  principles  of  sta- 
tistics must  be  of  doubtful  value. 

The  laws  of  statistics  are  applications  of  mathemat- 
ical formulas,  especially  of  permutations,  combinations 
and  probability.  Take  for  illustration  two  simple  laws 
(the  mathematical  derivation  of  them  would  not  be 
so  simple)  : 

1.  Suppose  a  number  of  measurements  have  been 
made.  If  the  measures  be  laid  of!  as  abscissas,  and 
the  number  of  times  each  measure  occurs  be  repre- 
sented graphically  as  the  corresponding  ordinate,  the 
line  drawn  through  the  points  thus  plotted  is  called 
the  distribution  *curve  for  these  measures.  The  area 
between  this  line  and  the  axis  of  x  is  the  surface  of 
frequency. 

If  a  quantity  one  is  measuring  is  due  to  chance  com- 
binations of  an  infinite  number  of  causes,  equal  in 
amount  and  independent,  and  all  equally  likely  to 
occur,  the  surface  of  frequency  is  of  the  form  shown 

156 


TREATMENT  OF  STATISTICS.  1 57 

in  Fig.  32,  the  equation  of  the  curve  being  y  =  e~~**. 
Most  effects  that  are  measured  are  not  due  to  such 
combinations  of  causes,  and  their  distribution  curves 
are  more  or  less  irregular ;  but  under  favorable  con- 
ditions they  frequently  approximate  this,  which  may 
be  called  the  normal,  "the  normal  probability  integral. " 
In  these  cases  the  tables  that  have  been  computed  for 
this  surface  are  of  great  assistance. 

2.  Every  one  knows  that,  other  things  being  equal,  the 
greater  the  number  of  measurements  made,  the  greater 
the  probability  of  their  average  (or  other  mean)  being 
the  true  one.     It  is  shown  mathematically  that  the 


probability  varies  as  the  square  root  of  the  number 
of  measures.  E.  g.,  If  in  one  investigation  64  cases 
were  measured,  and  in  another  25  cases,  the  returns 
from  the  first  investigation  will  be  more  trustworthy 
than  those  from  the  second  in  the  ratio  of  8  to  5. 

It  is  also  apparent  that,  if  the  average  deviation  (or 
other  measure  of  variability)  of  the  measures  from 
their  average  in  one  set  is  greater  than  in  another, 
the  average  is  less  trustworthy  in  that  set  in  which  the 
variability  is  the  greater.  Expressed  mathematically, 
the  trustworthiness  varies  inversely  as  the  variability. 
E.  g.,  in  one  investigation  the  average  deviation  of 


I58    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

the  measures  from  their  average  is  2  (2cm,  2  grams, 
or  whatever  the  unit  may  be)  while  in  another  in- 
vestigation (involving  the  same  number  of  measures 
etc.)  the  average  deviation  is  2.5.  Then  the  probable 
approach  to  accuracy  of  the  average  obtained  in  the 
first  investigation  is  to  that  of  the  average  obtained  in 
the  second  as  1/2  is  to  1/2.5,  or  as  5  to  4. 

If  the  two  investigations  differed  both  in  the  num- 
ber of  measures  and  in  the  deviation  from  the  average, 
both  would  enter  as  factors  in  determining  the  rela- 
tive confidence  to  be  reposed  in  the  two  results.  E.  g., 
combine  the  examples  in  the  two  preceding  para- 
graphs: An  average  was  obtained  from  64  measures 
whose  variability  was  2,  and  another  from  25  meas- 
ures whose  variability  was  2.5.     Then 

trustworthiness )  \  trustworthiness      |  /jrj     1       nz^       1 

of  first  average  )  '  of  second  average  F  "  '  -   2  "  2.5 

=  2:1 
The  trustworthiness  of  the  mean  of  a  number  of  meas- 
ures varies  directly  as  the  square  root  of  the  number 
of  measures  and  inversely  as  their  variability. 

The  foregoing  principles — the  A  B  C  of  statistical 
science — show  some  of  its  method  and  its  value  and 
the  direction  in  which  it  is  working.  Perhaps  the 
most  readable  treatise  on  the  subject  is  Professor 
Edward  L.  Thorndike's  Introduction  to  the  Theory 
of  Mental  and  Social  Measurements.  It  presupposes 
only  an  elementary  knowledge  of  mathematics  and 
contains  references  to  more  technical  works  on  the 
subject. 

Professor  W.  S  Hall,  in  "Evaluation  of  Anthropo- 
metric Data"  (Jour.  Am.  Med.  Assn.,  Chicago,  1901) 


TREATMENT  OF  STATISTICS.  I  59 

showed  that  the  curve  of  distribution  of  biologic  data 
is  the  curve  of  the  coefficients  in  the  expansion  of  an 
algebraic  binomial.  In  a  most  interesting  article,  "A 
Guide  to  the  Equitable  Grading  of  Students,"  in 
School  Science  and  Mathematics  for  June,  1906,  he 
applies  this  principle  to  the  distribution  of  student 
records  in  a  class. 

In  the  expansion  of  (a  +  &)5  there  are  6  terms,  and 
the  coefficients  are  1,  5,  10,  10,  5,  1.  Their  sum  is 
32.  If  320  students  do  their  work  and  are  tested  and 
graded  under  normal  (though  perhaps  unusual)  con- 
ditions, and  6  different  marks  are  used — say  A,  B,  C, 
D,  E,  F — the  number  of  pupils  attaining  each  of  these 
standings  should  approximate  10,  50,  100,  100,  50,  10, 
respectively.  If  3200  students  were  rated  in  6  groups 
under  similar  conditions,  the  numbers  in  the  groups 
would  be  ten  times  as  great — 100,  500,  1000  etc.,  and 
the  approximation  would  be  relatively  closer  than 
when  only  320  were  tested.  The  study  of  the  con- 
ditions that  cause  deviation  from  this  normal  distribu- 
tion of  standings  is  instructive  both  statistically  and 
pedagogically. 

A  rough-and-ready  statistical  method,  available  in 
certain  cases,  may  be  illustrated  as  follows :  Suppose 
we  are  engaged  in  ascertaining  the  number  of  words 
in  the  vocabulary  of  normal-school  juniors.  (Such 
an  investigation  is  now7  in  progress  under  the  direction 
of  Dr.  Margaret  K.  Smith,  of  the  normal  faculty  at 
New  Paltz.)  Let  us  select  at  random  a  page  of  the 
dictionary — say  the  13th — and  by  appropriate  tests 
ascertain  the  number  of  words  on  this  page  that  the 
pupil  knows,  divide  this  number  by  the  number  of 
words  on  the  page,  and  thus  obtain  a  convenient  ex- 


l60    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

pression  for  the  part  of  the  words  known.  Suppose 
the  quotient  to  be  .3016.  Turn  to  page  113  and  make 
similar  tests,  and  divide  the  number  known  on  both 
pages  by  the  number  of  words  on  both  pages,  giving 
— say —  .2391.  After  trying  page  213  the  result  is 
found  for  three  pages.  In  each  case  the  decimal 
represents  the  total  result  reached  thus  far  in  the 
experiment.     Suppose  the  successive  decimals  to  be 

.3016 
.2391 
.2742 
.2688 
.2562 
.2610 
.2628 
.2631 
.2642 
.2638 

A  few  decimals  thus  obtained  may  convince  the  ex- 
perimenter that  the  first  figure  has  "become  constant." 
Many  more  may  be  necessary  to  determine  the  second 
figure  unless  the  "series  converges"  rapidly  as  above. 
If  the  first  two  figures  be  found  to  be  26,  this  student 
knows  26%  of  the  words  in  the  dictionary.  Multi- 
plying the  "dictionary  total"  by  this  coefficient,  gives 
the  extent  of  the  student's  vocabulary,  correct  to  1% 
of  the  dictionary  total.  If  a  higher  degree  of  accuracy 
had  been  required,  a  three-place  coefficient  would  have 
been  determined. 

This  method  has  the  practical  advantage  that  the 
coefficient  found  at  each  step  furnishes,  by  comparison 
with  the  coefficients  previously  obtained,  an  indica- 
tion of  the  degree  of  accuracy  that  will  be  attained  by 


TREATMENT  OF  STATISTICS.  l6l 

its  use.  The  labor  of  division  may  be  diminished  by 
using,  on  each  page,  only  the  first  20  words  (or  other 
multiple  of  10).  Similarly  with  each  student  to  be 
examined.  The  method  here  described  is  applicable 
to  certain  classes  of  measures. 


MATHEMATICAL   SYMBOLS. 

The  origin  of  most  of  the  symbols  in  common  use 
may  be  learned  from  any  history  of  mathematics.  The 
noteworthy  thing  is  their  recentness.  Of  our  symbols 
of  operation  the  oldest  are  +  and  -,  which  appear  in 
Widmann's  arithmetic  (Leipsic,  1489). 

Consider  the  situation  in  respect  to  symbols  at  the 
middle  of  the  sixteenth  century.  The  radical  sign 
had  been  used  by  Rudolff,  (  ),  x,  ~-,  >  and  <  were 
still  many  years  in  the  future,  =  had  not  yet  appeared 
(though  another  symbol  for  the  same  had  been  used 
slightly)  and  +  and  -  were  not  in  general  use.  Almost 
everything  was  expressed  by  words  or  by  mere  ab- 
breviations. Yet  at  that  time  both  cubic  and  biquad- 
ratic equations  had  been  solved  and  the  methods  pub- 
lished. It  is  astonishing  that  men  with  the  intellectual 
acumen  necessary  to  invent  a  solution  of  equations  of 
the  third  or  fourth  degree  should  not  have  hit  upon 
a  device  so  simple  as  symbols  of  operation  for  the 
abridgment  of  their  work. 

The  inconvenience  of  the  lack  of  symbols  may  be 
easily  tested  by  writing — say — a  quadratic  equation 
and  solving  it  without  any  of  the  ordinary  symbols 
of  algebra. 

Even  after  the  introduction  of  symbols  began,  the 
process  was  slow.  But  recently  it  has  moved  with 
accelerating  velocity,  until  now  not  only  do  we  have 
a  symbol  for  each  operation — sometimes  a  choice  of 

1 6  j 


MATHEMATICAL  SYMBOLS.  163 

symbols — but  most  of  the  letters  of  the  alphabet  are 
engaged  for  special  mathematical  uses.     E.  g. : 

a  finite  quantity,   known   number,   side   of  triangle 

opposite  A,  intercept  on  axis  of  x>  altitude.  .  . 

b  known  number,  side  of  triangle  opposite  B,  base, 

intercept  on  axis  of  y.  .  . 
c  constant .  .  . 
d  differential,  distance... 
e  base  of  Napierian  logarithms. 

A  considerable  inroad  has  been  made  on  the  Greek 
alphabet,  e.  g. : 

y  inclination  to  axis  of  x. 

«•  3.14159... 

S  sum  of  terms  similarly  obtained. 

cr  standard  deviation  (in  theory  of  measurements). 
But  the  supply  of  alphabets  is  by  no  means  exhausted. 
There  is  no  cause  for  alarm. 


BEGINNINGS  OF  MATHEMATICS  ON  THE 

NILE. 

Whatever  the  excavations  in  Babylonia  and  Assyria 
may  ultimately  reveal  as  to  the  state  of  mathematical 
learning"  in  those  early  civilizations,  it  is  established 
that  in  Egypt  the  knowledge  of  certain  mathematical 
facts  and  processes  was  so  ancient  as  to  have  left  no 
record  of  its  origin. 

The  truth  of  the  Pythagorean  theorem  for  the  spe- 
cial case  of  the  isosceles  right  triangle  may  have  been 
widely  known  among  people  using  tile  floors  (see  Be- 
man  and  Smith's  New  Plane  Geometry,  p.  103).  That 
3,  4,  and  5  are  the  sides  of  a  right  triangle  was  known 
and  used  by  the  builders  of  the  pyramids  and  temples. 
The  Ahmes  papyrus  (1700  B.  C.  and  based  on  a  work 
of  perhaps  3000  B.  C.  or  earlier)  contains  many  arith- 
metical problems,  a  table  of  unit-fractions,  etc.,  and 
the  solution  of  simple  equations,  in  which  hau  (heap) 
represents  the  unknown.  Though  one  may  feel  sure 
that  arithmetic  must  be  the  oldest  member  of  the 
mathematical  family,  still  the  beginnings  of  arithmetic, 
algebra  and  geometry  are  all  prehistoric.  When  the 
curtain  raises  on  the  drama  of  human  history,  we  see 
men  computing,  solving  linear  equations,  and  using 
a  simple  case  of  the  Pythagorean  proposition. 


i64 


A  FEW  SURPRISING  FACTS  IN  THE  HIS- 
TORY OF  MATHEMATICS. 

That  spherical  trigonometry  was  developed  earlier 
than  plane  trigonometry  (explained  by  the  fact  that  the 
former  was  used  in  astronomy). 

That  the  solution  of  equations  of  the  third  and 
fourth  degree  preceded  the  use  of  most  of  the  symbols 
of  operation,  even  of  =. 

That  decimals — so  simple  and  convenient — should 
not  have  been  invented  till  after  so  much  "had  been 
attempted  in  physical  research  and  numbers  had  been 
so  deeply  pondered"  (Mark  Napier). 

That  logarithms  were  invented  before  exponents 
were  used ;  the  .derivation  of  logarithms  from  expo- 
nents—  now  always  used  in  teaching  logarithms — be- 
ing first  pointed  out  by  Euler  more  than  a  century 
later. 

That  the  earliest  systems  of  logarithms  (Napier's, 
Speidell's),  constructed  for  the  sole  object  of  facili- 
tating computation,  should  have  missed  that  mark 
(leaving  it  for  Briggs,  Gellibrand,  Vlacq,  Gunter  and 
others)  but  should  have  attained  theoretical  impor- 
tance, lending  themselves  to  the  purposes  of  modern 
analytical  methods  (Cajori). 


165 


QUOTATIONS  ON  MATHEMATICS. 

Following  are  some  of  the  quotations  that  have  been 
used  at  different  times  in  the  decoration  of  a  frieze 
above  the  blackboard  in  the  writer's  recitation  room: 

Let  no  one  who  is  unacquainted  with  geometry 
leave  here.  (This  near  the  door  and  on  the  inside — 
an  adaptation  of  the  motto  that  Plato  is  said  to  have 
had  over  the  outside  of  the  entrance  to  his  school  of 
philosophy,  the  Academy:  "Let  no  one  who  is  un- 
acquainted with  geometry  enter  here.,,) 

God  geometrizes  continually.    Plato. 

There  is  no  royal  road  to  geometry.    Euclid. 

Mathematics,  the  queen  of  the  sciences.     Gauss. 

Mathematics  is  the  glory  of  the  hurnan  mind.  Leib- 
nitz. 

Mathematics  is  the  most  marvelous  instrument  cre- 
ated by  the  genius  of  man  for  the  discovery  of  truth. 
Laisant. 

Mathematics  is  the  indispensable  instrument  of  all 
physical  research.     Berthelot. 

All  my  physics  is  nothing  else  than  geometry.  Des- 
cartes. 

There  is  nothing  so  prolific  in  utilities  as  abstrac- 
tions.    Faraday. 

The  two  eyes  of  exact  science  are  mathematics  and 
logic.    De  Morgan. 

All  scientific  education  which  does  not  commence 


166 


QUOTATIONS  ON    MATHEMATICS.  167 

with  mathematics  is,  of  necessity,  defective  at  its  foun- 
dation.    Compte. 

It  is  in  mathematics  we  ought  to  learn  the  general 
method  always  followed  by  the  human  mind  in  its 
positive  researches.     Compte. 

A  natural  science  is  a  science  only  in  so  far  as  it  is 
mathematical.    Kant. 

The  progress,  the  improvement  of  mathematics  are 
linked  to  the  prosperity  of  the  state.     Napoleon. 

If  the  Greeks  had  not  cultivated  conic  sections,  Kep- 
ler could  not  have  superseded  Ptolemy.     WhewcU. 

No  subject  loses  more  than  mathematics  by  any  at- 
tempt 'to  dissociate  it  from  its  history.     Glaisher. 


AUTOGRAPHS  OF  MATHEMATICIANS. 

For  the  photograph  from  which  this  cut  (Fig.  33) 
was  made  the  writer  is  indebted  to  Prof.  David  Eugene 
Smith.  x\s  an  explorer  in  the  bypaths  of  mathemat- 
ical history  and  a  collector  of  interesting  specimens 
therefrom,  Dr.  Smith  is,  perhaps,  without  a  peer.* 

The  reader  will  be  interested  to  see  a  facsimile  of 
the  handwriting  of  Euler  and  Johann  Bernoulli,  La- 
grange and  Laplace  and  Legendre,  Clifford  and  Dodg- 
son,  and  William  Rowan  Hamilton,  and  others  of  the 
immortals,  grouped  together  on  one  page.  In  the 
upper  right  corner  is  the  autograph  of  Moritz  Cantor, 
the  historian  of  mathematics.  On  the  sheet  overlap- 
ping that,  the  name  over  the  verses  is  faint ;  it  is  that 
of  J.  J.  Sylvester,  late  professor  in  Johns  Hopkins 
University. 

One  who  tries  to  decipher  some  of  these  documents 
may  feel  that  he  is  indeed  "In  the  Mazes  of  Mathe- 
matics."f  Mathematicians  are  not  as  a  class  noted 
for  the  elegance  or  the  legibility  of  their  chirography, 
and  these  examples  are  not  submitted  as  models  of 
penmanship.  But  each  bears  the  sign  manual  of  one 
of  the  builders  of  the  proud  structure  of  modern 
mathematics. 

*  Several  handsome  sets  of  portraits  of  mathematicians, 
edited  by  Dr.  Smith,  are  published  by  The  Open  Court  Pub- 
lishing Company. 

t  This  section  first  printed  in  a  series  bearing  that  title,  in 
The  Open  Court,  March — July,  1907. 

168 


AUTOGRAPHS  OF  MATHEMATICIANS. 


169 


bio 

E 


BRIDGES    AND    ISLES,    FIGURE    TRACING, 

UNICURSAL  SIGNATURES, 

LABYRINTHS. 


This  section  presents  a  few  of  the  more  elementary 
results  of  the  application  of  mathematical  methods  to 
these  interesting  puzzle  questions.* 


Fig.  34- 

The  city  of  Konigsberg  is  near  the  mouth  of  the 
Pregel  river,  which  has  at  that  point  an  island  called 
Kneiphof.  The  situation  of  the  seven  bridges  is  shown 
in  Fig.  34.     A  discussion  arose  as  to  whether  it  is 

*  For  more  extended  discussion,  and  for  proofs  of  the  theo- 
rems here  stated,  see  Euler's  Solutio  Problcmatis  ad  Geo- 
metriam  Situs  Per  tine  ntis,  Listing's  Vorstudicn  zur  Topologie, 

170 


BRIDGES  AND  ISLES,  LABYRINTHS  ETC.  171 

possible  to  cross  all  the  bridges  in  a  single  promenade 
without  crossing  any  bridge  a  second  time.  Euler's 
famous  memoir  was  presented  to  the  Academy  of 
Sciences  of  St.  Petersburg  in  1736  in  answer  to  this 
question.  Rather,  the  Konigsberg  problem  furnished 
him  the  occasion  to  solve  the  general  problem  of  any 
number  and  combination  of  isles  and  bridges. 

Conceive  the  isles  to  shrink  to  points,  and  the  prob- 
lem may  be  stated  more  conveniently  with  reference 
to  a  diagram  as  the  problem  of  tracing  a  given  figure 


Fig-  35- 

without  removing  the  pencil  from  the  paper  and  with- 
out retracing  any  part ;  or,  if  not  possible  to  .do  so 
with  one  stroke,  to  determine  hozv  many  such  strokes 
are  necessary.  Fig.  35  is  a  diagrammatic  represen- 
tation of  Fig.  34,  the  isle  Kneiphof  being  at  the 
point  K. 

The  number  of  lines  proceeding  from  any  point  of 
a  figure  may  be  called  the  order  of  that  point.     Every 

Ball's  Mathematical  Recreations  and  Essays,  Lucas's  Recrea- 
tions Mathematiques,  and  the  references  given  in  notes  by  the 
last  two  writers  named.  To  these  two  the  present  writer  is 
especially  indebted. 


172    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 


point  will  therefore  be  of  either  an  even  order  or  an 
odd  order.  E.  g.,  as  there  are  3  lines  from  point  A 
of  Fig.  36,  the  order  of  the  point  is  odd ;  the  order  of 
point  E  is  even.  The  well-known  conclusions  reached 
by  Enler  may  now  be  stated  as  follows : 


Fig.  36. 


Fig.  37-  Fig.  38. 

In  a  closed  figure  (one  with  no  free  point  or  "loose 
end")  the  number  of  points  of  odd  order  is  even, 
whether  the  figure  is  unicursal  or  not.  E.  g..  Fig.  36, 
a  multicnrsal  closed  figure,  has  four  points  of  odd 
order. 

A  figure  of  which  every  point  is  of  even  order  can 
be  traced  by  one  stroke  starting  from,  any  point  of  the 


BRIDGES  AND  ISLES,  LABYRINTHS  ETC. 


173 


figure.  E.g.,  Fig.  3/,  the  magic  pentagon,  symbol  of 
the  Pythagorean  school,  and  Fig.  38,  a  "magic  hexa- 
gram commonly  called  the  shield  of  David  and  fre- 
quently used  on  synagogues"  (Carus),  have  no  points 
of  odd  order ;  each  is  therefore  unicursal. 

A  figure  with  only  two  points  of  odd  order  can  be 
traced  by  one  stroke  by  starting  at  one  of  those  points. 
E.  g.,  Fig.  39  (taken  originally  from  Listing's  To  po- 
lo gie)  has  but  two  points  of  odd  order,  A  and  Z ;  it 
may  therefore  be  traced  by  one  stroke  beginning  at 
either  of  these  two  points  and  ending  at  the  other. 


Fig.  39- 


One  may  make  a  game  of  it  by  drawing  a  figure,  as 
Lucas  suggests,  like  Fig.  39,  but  in  larger  scale  on 
cardboard,  placing  a  small  counter  on  the  middle  of 
each  line  that  joins  two  neighboring  points,  and  setting 
the  problem  to  determine  the  course  to  follow  in  re- 
moving all  the  counters  successively  (simply  tracing 
continuously  and  removing  each  counter  as  it  is  passed, 
an  objective  method  of  recording  which  lines  have 
been  traced). 

A  figure  with  more  than  tzvo  points  of  odd  order 
is  multicnrsal.  E.  g..  Fig.  40  has  more  than  two  points 
of  odd  order  and  requires  more  than  one  course,  or 
stroke,  to  traverse  it. 


174    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 


Fig.  40. 


The  last  two  theorems  just  stated  are  special  cases 
of  Listing's: 

Let  Zn  represent  the  number  of  points  of  odd  order; 

then  n  strokes  are  neces- 
sary and  sufficient  to 
trace  the  figure.  E.  g., 
Fig.  39  with  2  points  of 
odd  order,  requires  one 
stroke;  Fig.  40,  repre- 
senting a  fragment  of 
masonry,  has  8  points  of  odd  order  and  requires  four 
strokes.  . 

Return  now  to  the  Konigsberg  problem  of  Fig.  34. 
By  reference  to  the  diagram  in  Fig.  35  it  is  seen  that 
there  are  four  points  of  odd  order.  Hence  it  is  not 
possible  to  cross  every  bridge  once  and  but  once  with- 
out taking  two  strolls. 

An  interesting  application  of  these  theorems  is  the 
consideration  of  the  number  of  strokes  necessary  to 
describe  an  n-gon  and  its  diagonals.  As  the  points 
of  intersection  of  the  diagonals  are  all  of  even  order, 
we  need  to  consider  only  the  vertexes.  Since  from 
each  vertex  there  is  a  line  to  every 
other  vertex,  the  number  of  lines 
from  each  vertex  is  n-l.  Hence, 
'if  n  is  odd,  every  point  is  of  even 
order,  and  the  entire  figure  can  be 
traced  unicursally  beginning  at  any 
point;  e.g.,  Fig.  41,  a  pentagon 
with  its  diagonals.  If  n  is  even, 
n  -  1  is  odd,  every  vertex  is  of  odd 
order,  the  number  of  points  of  odd 
order  is  n,  and  the  figure  can  not  be  described  in  less 


Fig.  4 


BRIDGES  AND  ISLES,  LABYRINTHS  ETC.  175 

than    n/2    courses ;    e.  g.,    Fig.    36,    quadrilateral,    re- 
quires two  strokes. 

Unicursal  signatures.  A  signature  (or  other  writ- 
ing) is  of  course  subject  to  the  same  laws  as  are  other 
figures  with  respect  to  the  number  of  times  the  pen 
must  be  put  to  the  paper.  Since  the  terminal  point 
could  have  been  connected  with  the  point  of  starting 
without  lifting  the  pen,  the  signature  may  be  counted 
as  a  closed  figure  if  it  has  no  free  end  but  these  two. 
The  number  of  points  of  odd  order  will  be  found  to 
be  even.     The  dot  over  an  i,  the  cross  of  a  t,  or  any 


Fig.  42.  Fig.  43. 

other  mark  leaving  a  free  point,  makes  the  signature 
multicursal.  There  are  so  many  names  not  requiring 
separate  strokes  that  one  would  expect  more  unicursal 
signatures  than  are  actually  found.  De  Morgan's  (as 
shown  in  the  cut  in  the  preceding  section)  is  one; 
but  most  of  the  signatures  there  shown  were  made 
with  several  strokes  each.  Of  the  signatures  to  the 
Declaration  of  Independence  there  is  not  one  that  is 
strictly  unicursal ;  though  that  of  Th  Jefferson  looks  as 
if  the  end  of  the  h  and  the  beginning  of  the  /  might 
often  have  been  completely  joined,  and  in  that  case 


I76    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

his   signature   would   have   been   written   in   a   single 
course  of  the  pen. 

Fig.  42,  formed  of  two  crescents,  is  "the  so-called 
sign  manual  of  Mohammed,  said  to  have  been  orig- 
inally traced  in  the  sand  by  the  point  of  his  scime.tar 
without  taking  the  scimetar  off  the  ground  or  re- 
tracing any  part  of  the  figure,"  which  can  easily  be 
done  beginning  at  any  point  of  the  figure,  as  it  con- 
tains no  point  of  odd  order.  The  mother  of  the 
writer   suggests   that,    if   the   horns   of   Mohammed's 


Fig.  44. 


crescents  be  omitted,  a  figure  (Fig.  43)  is  left  which 
can  not  be  traced  unicursally.  There  are  then  four 
points  of  odd.  order ;  hence  two  strokes  are  requisite 
to  describe  the  figure. 

Labyrinths  such  as  the  very  simple  one  shown  in 
Fig.  44  (published  in  1706  by  London  and  Wise) 
are  familiar,  as  drawings,  to  .every  one.  In  some  of 
the  more  complicated  mazes  it  is  not  so  easy  to  thread 
one's   way,    even    in    the    drawing,    where   the    entire 


BRIDGES  AND  ISLES,  LABYRINTHS  ETC.  1 77 

maze  is  in  sight,  while  in  the  actual  labyrinth,  where 
walls  or  hedges  conceal  everything  but  the  path  one 
is  taking  at  the  moment,  the  difficulty  is  greatly  in- 
creased and  one  needs  a  rule  of  procedure. 

The  mathematical  principles  involved  are  the  same 
as  for  tracing  other  figures  ;  but  in  their  application 
several  differences  are  to  be  noticed  in  the  conditions 
of  the  two  problems.  A  labyrinth,  as  it  stands,  is  not 
a  closed  figure ;  for  the  entrance  and  the  center  are 
free  ends,  as  are  also  the  ends  of  any  blind  alleys  that 
the  maze  may  contain.  These  are  therefore  points  of 
odd  order.  There  are  usually  other  points  of  odd 
order.  Hence  in  a  single  trip  the  maze  can  not  be 
completely  traversed.  But  it  is  not  required  to  do  so. 
The  problem  here  is,  to  go  from  the  entrance  to  the 
center,  the  shorter  the  route  found  the  better.  More- 
over, the  rules  of  the  game  do  not  forbid  retracing 
one's  course. 

It  is  readily  seen  (as  first  suggested  by  Euler)  that 
by  going  over  each  line  twice  the  maze  becomes  a 
closed  figure,  terminating  where  it  begins,  at  the  en- 
trance, including  the  center  as  one  point  in  the  course, 
and  containing  only  points  of  even  order.  Hence 
every  labyrinth  can  be  completely  traversed  by  going 
over  every  path  twice — once  in  each  direction.  It  is 
only  necessary  to  have  some  means  of  marking  the 
routes  already  taken  (and  their  direction)  to  avoid 
the  possibility  of  losing  one's  way.  This  duplication 
of  the  entire  course  permits  no  failure  and  is  so  general 
a  method  that  one  does  not  need  to  know  anything 
about  the  particular  labyrinth  in  order  to  traverse  it 
successfully  and  confidently.  But  if  a  plan  of  the 
labyrinth  can  be  had,  a  course  may  be  found  that  is 
shorter. 


178    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

Theseus,  as  he  threaded  the  Cretan  labyrinth  in  quest 
of  the  Minotaur,  would  have  regarded  Euler's  mathe- 
matical theory  of  mazes  as  much  less  romantic  than 
the  silken  cord  with  Ariadne  at  the  outer  end ;  but 
there  are  occasions  where  a  modern  finds  it  necessary 
to  "go  by  the  book."  Doubtless  the  labyrinth  of 
Daedalus  was  "a  mighty  maze,  but  not  without  a  plan."" 

Fig.  45  presents  one  of  the  most  famous  labyrinths, 
though  by  no  means  among  the  most  puzzling.  It  is 
described  in  the  Eneyclop&dia  Britannica  (article 
"Labyrinth")  as  follows: 


Fig.  45- 


"The  maze  in  the  gardens  at  Hampton  Court  Pal- 
ace is  considered  to  be  one  of  the  finest  examples  in 
England.  It  was  planted  in  the  early  part  of  the  reign 
of  William  III,  though  it  has  been  supposed  that  a 
maze  had  existed  there  since  the  time  of  Henry  VIII. 
It  is  constructed  on  the  hedge  and  alley  system,  and 
was,  we  believe,  originally  planted  with  hornbeam, 
but  many  of  the  plants  have  died  out,  and  been  re- 
placed by  hollies,  yews,  etc.,  so  that  the  vegetation  is 
mixed.  The  walks  are  about  half  a  mile  in  length, 
and  the  extent  of  ground  'occupied  is  a  little  over  a 
quarter  of  an  acre.     The  center  contains  two  large 


BRIDGES  AND  ISLES,  LABYRINTHS  ETC.  1 79 

trees,  with  a  seat  beneath  each.  The  key  to  reach  this 
resting  place  is  to  keep  the  right  hand  continuously  in 
contact  with  the  hedge  from  first  to  last,  going  round 
all  the  stops." 


THE  NUMBER  OF  THE  BEAST. 

"Here  is  wisdom.  He  that  hath  understanding,  let 
him  count  the  number  of  the  beast ;  for  it  is  the  num- 
ber of  a  man :  and  his  number  is  Six  hundred  and  sixty 
and  six."  (Margin,  "Some  ancient  authorities  read 
Six  hundred  and  sixteen/')     Revelation   13:18. 

No  wonder  that  these  words  have  been  a  powerful 
incentive  to  a  class  of  interpreters  who  delight  in 
apocalyptic  literature,  especially  to  such  as  have  a 
Pythagorean  regard  for  hidden  meaning  in  numbers. 

There  were  centuries  in  which  no  satisfactory  inter- 
pretation was  generally  known.  At  about  the  same 
time,  in  1835,  Benary,  Fritzsche,  Hitzig  and  Reuss 
connected  the  number  666  with  "Emperor  (Csesar) 
Neron,"  pn:iDp.  In  the  number  notation  of  the  He- 
brews the  letter  p  =  100,  D  =  60,  -)  =  200,  3=50,  ^  =  200, 
1=6,  *]  =  50.  These  numbers  added  give  666.  Omit- 
ting the  final  letter  from  the  name  (making  it  "Em- 
peror Nero")  the  number  represented  is  616,  the  mar- 
ginal reading.  The  present  writer's  casual  opinion 
is  that  the  foregoing  is  the  meaning  intended  in  the 
passage ;  and  that  after  the  fear  of  Nero  passed,  the 
knowledge  of  the  meaning  of  the  number  gradually 
faded,  and  had  to  be  rediscovered  long  afterward. 
It  is,  however,  strange,  that  only  about  a  century  after 
the  writing  of  the  Apocalypse  the  connection  of  the 
number  with  Nero  was  apparently  unknown  to  Ire- 


180 


THE  NUMBER  OF  THE  BEAST.  l8l 

naeus.  He  made  several  conjectures  of  words  to  fit 
the  number. 

In  the  later  Middle  Ages  and  afterward,  the  num- 
ber was  made  to  fit  heresies  and  individual  heretics. 
Protestants  in  turn  found  that  a  little  ingenuity  could 
discover  a  similar  correspondence  between  the  num- 
ber and  symbols  for  the  papacy  or  names  of  popes. 
So  the  exchange  of  these  expressions  of  regard  con- 
tinued. When  the  name  is  taken  in  Greek,  the  number 
is  expressed  in  Greek  numerals,  where  every  letter  is 
a  numeral ;  but  when  Latin  is  used,  only  M,  D,  C,  L,  X, 
V  and  I  have  numerical  values. 

VICARIVS     FILII     DEI 

5  +1+100  +1  +5  +1+50+1  +1+500      +1  =  666 

This  and  a  similar  derivation  from  Luther's  name  are 
perhaps  the  most  famous  of  these  performances. 

De  Morgan  cites  a  book  by  Rev.  David  Thorn,* 
from  which  he  quotes  names,  significant  mottoes  etc. 
that  have  been  shown  to  spell  out  the  number  666. 
He  gives  18  such  from  the  Latin  and  38  from  the 
Greek  and  omits  those  from  the  Hebrew.  Some  of 
these  were  made  in  jest,  but  many  in  grim  earnest. 
He  also  gives  a  few  from  other  sources  than  the  book 
mentioned. 

The  number  of  such  interpretations  is  so  great  as 
to  destroy  the  claim  of  any.  "We  can  not  infer  much 
from  the  fact  that  the  key  fits  the  lock,  if  it  is  a  lock 
in  which  almost  any  key  will  turn."  A  certain  interest 
still  attaches  to  all  such  cabalistic  hermeneutics,  and 
they  are  not  without  their  lesson  to  us,  but  it  is  not  the 
lesson  intended  by  the  interpreter.     When  it  comes  to 

*  The  Number  and  Names  of  the  Apocalyptic  Beasts,  part 
I,  Svo,  1848.     See  De  Morgan's  Budget  of  Paradoxes,  p.  402-3 


1 82    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

the  use  of  such  interpretations  by  one  branch  of  the 
Church  against  another,  one  would  prefer  as  less  ir- 
reverent the  suggestion  of  De  Morgan,  that  the  true 
explanation  of  the  three  sixes  is  that  the  interpreters 
are  "six  of  one  and  half  a  dozen  of  the  other." 


MAGIC  SQUARES. 


"A  magic  square  is  one  divided  into  any  number 
of  equal  squares,  like  a  chess-board,  in  each  of  which 
is  placed  one  of  a  series  of  consecutive  numbers  from 
1  up  to  the  square  of  the  number  of  cells  in  a  side, 
in  such  a  manner  that  the  sum  of  those  in  the  same  row 
or  column  and  in  each  of  the  two  diagonals  is  con- 
stant/'.   (Encyclopedia  Britannica.) 

The  term  is  often  extended  to  include  an  assemblage 
of  numbers  not  consecutive  but  meeting  all  other  re- 
quirements of  this  definition.  If  every  number  in  a 
magic  square  be  multiplied  by  any  number,  q,  integral 
or  fractional,  arithmetical,  real  or  imaginary,  such 
an  assemblage  is  formed,  and  by  the  distributive  law 
of  multiplication  its  sums  are 
each  q  times  those  in  the  orig- 
inal square. 

One  way  (DelaLoubere's) 
of  constructing  any  odd-num- 
ber square  is  as  follows :  r  y 

1.  In  assigning  the  consecu- 
tive  numbers,   proceed   in   an 
oblique    direction    up    and    to 
the  right  (see  4,  5,  6  in  Fig.  p;g-  ^ 
46). 

2.  When  this  would  carry  a  number  out  of  the 
square,  write  that  number  in  the  cell  at  the  opposite 
end  of  the  column  or  row,  as  shown  in  case  of  the 
canceled  figures  in  the  margin  of  Fig.  46. 

183 


X  i  i 


8 

1 

6 

3 

5 

T 

4 

9 

2 

184    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

3.  When  the  application  of  rule  1  would  place  a 
number  in  a  cell  already  occupied,  write  the  new  num- 
ber, instead,  in  the  cell  beneath  the  one  last  filled. 
(The  cell  above  and  to  the  right  of  3  being  occupied, 
4  is  written  beneath  3.) 

4.  Treat  the  marginal  square  marked  x  as  an  oc- 
cupied cell,  and  apply  rule  3. 

5.  Begin  by  putting  1  in  the  top  cell  of  the  middle 
column. 

This  rule  will  fill  any  square  having  an  odd  number 
of  cells  in  each  row  and  column. 

The  investigation  of  some  of  the  properties  of  the 
simple  squares  just  described  is  an  interesting  diver- 
sion. For  example,  after  the  5-square  and  7-square 
have  been  constructed  and  one  is  familiar  with  the  rule, 
he  may  set  himself  the  problem  to  find  a  formula  for 
the  sum  of  the  numbers  in  each  row,  column  or  diagonal 
of  any  square.  Noticing  that  the  diagonal  from  lower 
left  corner  to  upper  right  is  composed  of  consecutive 
numbers,  it  will  be  easy  to  write  the  formula  for  the  sum 
of  that  series  (the  required  sum)  if  we  can  find  the  form- 
ula for  the  number  in  the  lower  left  corner.  Since  the 
number  of  cells  in  each  row  or  column  of.  the  squares 
we  are  considering  is  odd,  we  represent  that  number 
by  the  general  formula  for  an  odd  number,  2w+l. 
Our  square,  then,  is  a  (2/H-l) -square.  If  n  be  taken 
=  1,  we  have  a  3-square ;  if  »  =  2,  a  5-square;  etc. 
Now  it  is  seen  by  inspection  that  the  number  in  the 
lower  left  cell  is  n(2n+l)+l,  the  succeeding  numbers 
in  the  diagonal  being  n(2n+l)  +  2,  »(2fH-i)+3,  etc. 
Summing  this  series  to  2/2+1  terms,  we  have  the  re- 
quired formula,  (2n+l)  (2n2+2n+l).  This  might  be 
tabulated  as  follows  (including  1  as  the  limiting  case 
of  a  magic  square)  : 


MAGIC  SQUARES. 


185 


arbitrary 
values  of 
n  (succes- 
S  I  V  e   in- 
tegers) 

NO.OF  CELLS 
IN     EACH 
ROW       OR 
COLUMN 

(succes- 
sive ODD 
numbers) 

THE  NUMBER  IN 
THE    LOWER 
LEFT    CORNER 

SUM   OF  THE    NUMBERS    IN 
ANY  ROW,    COLUMN  OR  DI- 
AGONAL 

n 

2«+i 

rc(2»+l)+l 

(2*4-1)  (2»2+2*+l) 

0 

1 

1 

1 

1 

3 

4 

15 

2 

5 

11 

65 

3 

7 

22 

175 

4 

9 

37 

369 

5 

11 

56 

671 

etc. 

etc. 

Following  is  the  11-square 

68  81  94  107  120  1 

80  93  106  119  11-  13 

92  105  118  10  12  25 

104  117   9  22  24  37 

116   8  21  23     36  49 

7  20  33     35  48  61 

19  32  34  47  60  73 

31  44  46  59  .72  85 

43  45  58  71  84  97 

55  57  70  83    96  109 

56  69  82  95  108  121 


sum,  671 : 

14 

27 

40 

53 

66 

26 

39 

52 

65 

67 

38 

51 

64 

77 

79 

50 

63 

76 

78 

91 

62 

75 

88 

90 

103 

74 

87 

89 

102 

115 

86 

99 

101 

114 

6 

98 

100 

113 

5 

18 

110 

112 

4 

17 

30 

111 

3 

16 

29 

42 

2 

15 

28 

41 

54 

l86    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 


256 

2 

64 

8 

32 

128 

16 

512 

4 

Fig.  47- 


There  are  also  "geometrical  magic  squares,"  in 
which  the  product  of  the  numbers  in  every  row,  col- 
umn and  diagonal  is  the  same.  If  a  number  be  selected 
as  base  and  the  numbers  in  an  ordinary  magic  square 
be  used  as  exponents  by  which  to  affect  it,  the  result- 
ing powers  form  a  geometric  square  (by  the  first  law 
of  exponents).  E.g.,  Take  2  as  base  and  the  num- 
bers in  the  square  (Fig.  46)  as 
exponents.  The  resulting  geo- 
metrical magic  square  (Fig.  47) 
has  215  for  the  product  of  the 
numbers  in  each  line. 

The  theory  of  magic  squares 
in  general,  including  even-num- 
ber squares,  squares  with  addi- 
tional .properties,  etc.,  and  inclu- 
ding the  extension  of  the  idea  to 
cubes,  is  given  in  the  article  "Magic  Squares"  in  the 
Encyclopedia  Britannica,  together  with  some  account 
of  their  history.  See  also  Ball's  Recreations ;  Lucas's 
Recreations,  vol.  4,  Cinquieme  Recreation,  "Les  Car- 
res magiques  de  Fermat" ;  and  the  comprehensive 
article,  "A  Mathematical  Study  of  Magic  Squares," 
by  L.  S.  Frierson,  in  The  Monist  for  April,  1907,  p. 
272-293. 

The  oldest  manuscript  on  magic  squares  still  pre- 
served dates  from  the  fourth  or  fifth  century.  It  is 
by  a  Greek  named  Moscopulus.  Magic  squares  en- 
graved on  metal  or  stone  are  said  to  be  worn  as  talis- 
mans in  some  parts  of  India  to  this  day.  (Britaiinica.) 
Among  the  most  prominent  of  the  modern  philos- 
ophers who  have  amused  themselves  by  perfecting 
the  theory  of  magic  squares  is  Franklin,  "the  model 
of  practical  wisdom." 


MAGIC  SQUARES. 


187 


Domino  magic  squares.  A  pleasing  diversion  is  the 
forming  of  magic  squares  with  dominoes.  This  phase 
of  the  subject  has  been  set  forth  by  several  writers ; 
among  them  Ball,*  who  also  mentions  coin  magic 
squares.     The  following  are  by  Mr.  Escott,  who  re- 


Fig.  48. 


Fig.  49. 


marks :  "I  do  not  know  how  many  solutions  there  are. 
I  give  five  [of  which  two  are  reproduced  here],  which 
I  found  after  a  few  trials.     In  each  of  these  magic 
squares  the   sum   is  the 
greatest  possible,  19.    If 
we  subtract  every  num- 
ber from  6,  we  get  magic 
squares  where  the  sum  is 
the  least  possible,  5." 

Magic  h exagon s. f 

Sum  of  any  side  of  tri- 
angle =  sum  of  vertexes 
of  either  triangle = sum  of 
vertexes  of  convex  hexa- 
gon =  sum  of  vertexes  of 

any  parallelogram  =  26.    'There  are  only  six  solutions, 
of  which  this  is  one."     (Fig.  50.) 

*  Recreations,  p.  165-6. 

f  From  Mr.  Escott,  who  says:  "The  first  appeared  in  Knowl- 
edge, in  1895.  and  the  second  is  due  to  Mr.  S.  Lloyd." 


l88    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

Place  the  numbers  1  to  19  on  the  sides  of  the  equi- 
lateral triangles  so  that  the  sum  on  every  side  is  the 
same. 


8 a3  A 1-9 a3 


\\        i  o        p        fj-  jg         ie        u        12 

[ 1-3 V 1-6—44         4-A— 1-3— -V 9- 

\        M       H        /IO  *?        J  5        7 

ii 


9^8  2  io 

Fig.  5i.  Fig.  52. 

The  sum  on  the  sides  of  the  triangles  in  Fig.  51 
is  22.  In  Fig.  52  it  is  23.  If  we  subtract  each  of  the 
above  numbers  from  20,  we  have  solutions  where  the 
sums  are  respectively  38  and  37, 


THE   SQUARE   OF  GOTHAM. 

(From  Teachers  Note  Book,  by  permission.) 


The  wise  men  of  Gotham,  famous  for  their  eccentric 
blunders,  once  undertook  the  management  of  a  school ; 
they  arranged  their  establishment  in  the  form  of  a 
square  divided  into  9  rooms.  The  playground  occu- 
pied the  center,  and  24  scholars  the  rooms  around  it, 
3  being  in  each.  In  spite  of  the  strictness  of  discipline, 
it  was  suspected  that  the  boys  were  in  the  habit  of 


3         3         3 
3  3 

3         3         3 


4 

» 

4 

i 

I 

4 

I 

4 

Fig.  53- 


Fig.  54- 


playing  truant,  and  it  was  determined  to  set  a  strict 
watch.  To  assure  themselves  that  all  the  boys  were 
on  the  premises,  they  visited  the  rooms,  and  found  3 
in  each,  or  9  in  a  row.  Four  boys  then  went  out,  and 
the  wise  men  soon  after  visited  the  rooms,  and  finding 
9  in  each  row,  thought  all  was  right.  The  four  boys 
then  came  back,  accompanied  by  four  strangers ;  and 
the  Gothamites,  on  their  third  round,  finding  still  9 
in  each  row,  entertained  no  suspicion  of  what  had 
taken  place.     Then  4  more  "churns"  were  admitted, 

189 


I90    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 


but  the  wise  men,  on  examining  the  establishment  a 
fourth  time,  still  found  9  in  each  row,  and  so  came 
to  an  opinion  that  their  previous  suspicions  had  been 
unfounded. 

Figures  53-56  show  how  all  this  was  possible,  as  they 


2 

5 

2 

5 

5 

2 

5 

2 

1 

7 

1 

7 

7 

1 

7 

1 

Fig.  55- 


Fig.  56. 


represent  the  contents  of  each  room  at  the  four  dif- 
ferent visits;  Fig.  53,  at  the  commencement  of  the 
watch;  Fig.  54,  when  four  had  gone  out;  Fig.  55, 
when  the  four,  accompanied  by  another  four  had  re- 
turned ;  and  Fig.  56,  when  four  more  had  joined  them. 


A  MATHEMATICAL  GAME-PUZZLE. 

"Place  15  checkers  on  the  table.  You  are  to  draw 
(take  away  either  1,  2  or  3)  ;  then  your  opponent  is  to 
draw  (take  1,  2  or  3  at  his  option)  ;  then  you  draw 
again ;  then  your  opponent.  You  are  to  force  him  to 
take  the  last  one." 

Solution :  When  your  opponent  makes  his  last  draw, 
there  must  be  just  one  checker  left  for  him  to  take. 
Since  at  every  draw  you  are  limited  to  removing  either 
1,  2  or  3,  you  can,  by  your  last  draw,  leave  just  1  if, 
and  only  if,  you  find  on  the  board  before  that  draw 
either  2,  3  or  4.  You  must,  therefore,  after  your  next 
to  the  last  draw,  leave  the  board  so  that  he  can  not 
but  leave,  after  his  next  to  the  last  draw,  either  2,  3  or 
4.  5  is  clearly  the  number  that  you  must  leave  at  that 
time;  since  if  he  takes  1,  he  leaves  4;  if  2,  3;  if  3,  2. 
Similarly,  after  your  next  preceding  draw  you  must 
leave  9 ;  after  your  next  preceding,  13 ;  that  is,  yon 
must  first  draw  2.  Then  after  each  draw  that  he  makes, 
you  draw  the  difference  between  4  and  the  number 
that  he  has  just  drazvn,  (if  he  takes  1,  you  follow  by 
taking  3  ;  if  he  takes  2,  you  take  2 ;  if  he  takes  3,  you 
take  1).  Four  being  the  sum  of  the  smallest  number 
and  the  largest  that  may  be  drawn,  you  can  always 
make  the  sum  of  two  consecutive  draws  (your  op- 
ponent's and  yours)  4,  and  you  can  not  always  make  it 
any  other  number. 

Following  would  be  a  more  general  problem:  Let 

191 


192    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

your  opponent  place  on  the  board  any  number  of 
checkers  leaving  you  to  choose  who  shall  first  draw 
(1,  2  or  3  as  before).  Required  to  leave  the  last 
checker  for  him.  Solution:  If  the  number  he  places 
on  the  board  is  a  number  of  the  form  4n+  1,  choose 
that  he  shall  draw  first.  Then  keep  the  number  left 
on  the  board  in  that  form  by  making  his  draw  +  yours 
=  4,  until  n  =  0;  that  is,  until  there  is  but  one  left.  If 
the  number  that  he  places  on  the  board  is  not  of  that 
form;,  draw  first  and  reduce  it  to  a  number  that  is  of 
that  form,  and  proceed  as  before. 

The  problem  might  be  further  generalized  by  vary- 
ing the  number  that  may  be  taken  at  a  draw. 


PUZZLE  OF  THE  CAMELS. 

There  was  once  an  Arab  who  had  three  sons.  In 
his  will  he  bequeathed  his  property,  consisting  of 
camels,  to  his  sons,  the  eldest  son  to  have  one-half  of 
them,  the  second  son  one-third,  and  the  youngest  one- 
ninth.  The  Arab  died  leaving  17  camels,  a  number 
not  divisible  by  either  2,  3  or  9.  As  the  camels  could 
not  be  divided,  a  neighboring  sheik  was  called  in  con- 
sultation. 

He  loaned  them  a  camel,  so  that  -they  had  18  to  di- 
vide. 

The  first  son  took  1/2 9 

The  second  took  1/3 .     .     6 

The  third  took '*/■ _2 

Total 17 

They  had  divided  equitably,  and  were  able  to  return 
the  camel  that  had  been  loaned  to  them. 

It  should  be  noted  that  l/2  +  .*/s .+ 1/9  =  17/18, -"not 
unity.     The  numbers  9,  6,  2  are  in  the  same  ratio  as 

7,>,  Y„  V.. 

This  is  probably  an  imitation  of  the  old  Roman 
inheritance  problem  which  may  be  found  in  Cajori's 
History  of  Mathematics,  p.  79-80,  or  in  his  History 
of  Elementary  Mathematics,  p.  41. 


193 


A  FEW  MORE  OLD-TIMERS. 

A  man  had  eight  gallons  of  wine  in  a  keg.  He 
wanted  to  divide  it  so  as  to  get  one-half.  He  had  no 
measures  but  a  three  gallon  keg,  a  five  gallon  keg  and 
a  seven  gallon  keg.  How  did  he  divide  it?  (The 
five  gallon  keg  is  unnecessary.) 

Only  one  dimension  on  Wall  street.  Broker  (de- 
termined to  see  the  bright  side)  :  "Every  time  I 
bought  stocks  for  a  rise,  they  went  down ;  and  when 

1  sold  them,  they  went  up.  Luckily  they  can't  go 
sidewise." 

The  apple  women.  Two  apple  women  had  30  apples 
each  for  sale.     If  the  first  had  sold  hers  at  the  rate  of 

2  for  1  cent,  she  would  have  received  15  cents.  If  the 
other  had  sold  hers  at  3  for  1  cent,  she  would  have 
received  10  cents.  Both  would  have  had  25  cents. 
But  they  put  them  all  together  and  sold  the  60  apples 
a*  5  for  2  cents,  thus  getting  24  cents.  What  became 
of  the  other  cent? 

G.  D.  with  same  remainder.  Given  three  (or  more) 
integers,  as  27,  48,  90;  required  to  find  their  greatest 
integral  divisor  that  will  leave  the  same  remainder. 

Solution :  Subtract  the  smallest  number  from  each 
of  the  others.  The  G.  C.  D.  of  the  differences  is  the 
required  divisor.     48-27  =  21;  90-27  =  63;  G.  C.  D. 

194 


A  FEW  MORE  OLD  TIMERS.  I95 

of  21  and  63  is  21.     If  the  given  numbers  be  divided 
by  21,  there  is  a  remainder  of  6  in  each  case. 

"15  Christians  and  15  Turks,  being  at  sea  in  one  and 
the  same  ship  in  a  terrible  storm,  and  the  pilot  de- 
claring a  necessity  of  casting  the  one  half  of  those 
persons  into  the  sea,  that  the  rest  might  be  saved ; 
they  all  agreed,  that  the  persons  to  be  cast  away  should 
be  set  out  by  lot  after  this  manner,  viz.,  the  30  persons 
should  be  placed  in  a  round  form  like  a  ring,  and 
then  beginning  to  count  at  one  of  the  passengers, 
and  proceeding  circularly,  every  ninth  person  should 
be  cast  into  the  sea,  until  of  the  30  persons  there  re- 
mained only  15.  The  question  is,  how  those  30  persons 
should  be  placed,  that  the  lot  might  infallibly  fall 
upon  the  15  Turks  and  not  upon  any  of  the  15  Chris- 
tians." 

The  early  history  of  this  problem  is  given  by  Pro- 
fessor Cajori  in  his  History  of  Elementary  Mathe- 
matics, p.  221-2,  who  also  quotes  mnemonic  verses 
giving  the  solution:  4  Christians,  then  5  Turks,  then 
2  Christians,  etc. 

The  solution  is  really  found  by  arranging  30  num- 
bers or  counters  in  a  ring,  or  in  a  row  to  be  read 
in  circular  order.  Count  according  to  the  conditions 
of  the  problem,  marking  every  ninth  one  "T"  until 
15  are  marked,  then  mark  the  remaining  15  "C." 

The  same  problem  has  appeared  in  other  forms. 
Sometimes  other  classes  of  persons  take  the  places  of 
the  Christians  and  Turks,  sometimes  every  tenth  one 
is  lost  instead  of  every  ninth. 


A  FEW  CATCH  QUESTIONS. 

What  number  can  be  divided  by  every  other  number 
without  a  remainder? 

"Four- fourths  exceeds  three-fourths  by  what  frac- 
tional part?"  This  question  will  usually  divide  a  com- 
pany. 

Can  a  fraction  whose  numerator  is  less  than  its  de- 
nominator be  equal  to  a  fraction  whose  numerator 
is   greater   than   its   denominator?      If   not,   how   can 

-3=   +5 
+  6      -10- 
In  the  proportion 

+  6:-3  ::-10  :  +  5 
is  not  either  extreme  greater  than  either  mean  ?   What 
has  become  of  the  old  rule,  "greater  is  to  less  as  greater 
is  to  less"? 

Where  is  the  fallacy  in  the  following? 
1  mile  square   =  1  square  mile 
.-.2  miles  square  -2  square  miles   (Axiom:   If 
equals  be  multiplied  by  equals,  etc.) 
Or  in  this  (which  is  from  Rebiere)  : 
A  glass   l/2   full  of  water  =  a  glass  y2  empty 

.-.  A  glass  full  =  a  glass  empty  (Axiom: 
If  equals  be  multiplied.) 

196 


SEVEN-COUNTERS   GAME. 

Required  to  place  seven  counters  on  seven  of  the 
eight  spots  in  conformity  to  the  following  rule :  To 
place  a  counter,  one  must  set  out  from  a  spot  that  is 
unoccupied  and  move  along  a  straight  line  to  the  spot 
where  the  counter  is  to  be  placed. 


The  writer  remembers  seeing  this  as  a  child  when 
the  game  was  probably  new.  The  solution  is  so  easy 
that  it  offered  no  difficulty  then.  A  puzzle  whose  so- 
lution is  seen  by  almost  any  one  in  a  minute  or  two 
is  hardly  worth  a  name,  and  one  wonders  to  see  it  in 
Lucas's  Recreations  mathematiques  and  dignified  by 
the  title  "The  American  Game  of  Seven  and  Eight." 
Lucas  explains  that  the  game,  invented  by  Knowlton, 

197 


I98    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

of  Buffalo,  N.  Y.,  was  published,  in  1883,  by  an  Amer- 
ican journal  offering  at  first  a  prize  to  the  person  who 
should  send,  within  a  fixed  time,  the  solution  expressed 
in  the  fewest  words. 

Lucas's  statement  of  the  solution  is,  Take  always 
for  point  of  destination  the  preceding  point  of  de- 
parture. Starting,  for  example,  from  the  point  4,  fol- 
lowing the  line  4—1,  and  placing  a  counter  at  1  ;  the 
spot  4  must  be  the  second  spot  of  arrival.  As  one  can 
reach  4  only  by  the  line  7-4,  the  spot  7  will  necessarily 
be  the  second  point  of  departure ;  etc.,  the  seven  moves 
being 

4-1,  7-4,  2-7,  5-2,  8-5,  3-8,  6-3. 

Lucas*  generalizes  the  game  somewhat  and  adds 
other  amusements  with  counters,  less  trivial  than  "the 
American  game." 

*  Vol.  3,  sixth  recreation,  from  which  the  figure  and  descrip- 
tion in  the  text  are  taken. 


TO  DETERMINE  DIRECTION  BY  A  WATCH. 

Those  who  are  familiar  with  this  very  elementary 
operation  usually  take  it  for  granted  that  every  one 
knows  it.  Inquiry  made  recently  in  a  class  of  normal 
school  students  revealed  the  fact  that  but  few  had 
heard  of  it  and  not  one  could  explain  or  state  the 
method.  The  writer  has  not  infrequently  known  well 
informed  persons  to  express  surprise  and  pleasure  at 
hearing  it. 

With  the  face  of  the  watch  up,  point  the  hour  hand 
to  the  sun.  Then  the  point  midway  between  the  pres- 
ent hour  mark  and  XII  is  toward  the  south.  E.  g.,  at 
4  o'clock,  when  the  hour  hand  is  held  toward  the  sun, 
II  is  toward  the  south. 

Or  the  rule  may  be  stated  thus :  With  the  point  that 
is  midway  between  the  present  hour  and  XII  held 
toward  the  sun,  XII  is  toward  the  south.  E.  g.,  at 
4  o'clock  hold  II  toward  the  sun ;  then  the  line  from 
the  center  of  the  face  to  the  mark  XII  is  the  south 
line. 

The  reason  is  apparent.  At  12  o'clock  the  sun,  the 
hour  hand  and  the  XII  mark  are  all  toward  the  south. 
The  sun  and  the  hour  hand  revolve  in  the  same  direc- 
tion, but  the  hour  hand  makes  the  complete  revolution 
in  12  hours,  the  sun  in  24.     Hence  the  rule. 

The  errors  due  to  holding  the  watch  horizontal  in- 
stead of  in  the  plane  of  the  ecliptic,  and  to  the  differ- 


199 


200    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

ence  between  standard  time  and  solar  time,  are  negli- 
gible for  the  purpose  to  which  this  rule  is  usually  put. 

Ball*  mentions  that  the  "rule  is  given  by  W.  H. 
Richards,  Military  Topography,  London,  1883."  Be- 
ing so  simple  and  convenient,  it  was  probably  known 
earlier. 

Professor  Ball  also  gives  (p.  356)  the  rule  for  the 
southern  hemisphere:  "If  the  watch  is  held  so  that 
the  figure  XII  points  to  the  sun,  then  the  direction 
which  bisects  the  angle  between  the  hour  of  the  day 
and  the  figure  XII  will  point  due  north." 

*  Recreations,  p.  355. 


MATHEMATICAL  ADVICE  TO  A  BUILDING 
COMMITTEE. 

It  will  be  remembered  that  the  man  who,  under  the 
pseudonym  Lewis  Carroll,  wrote  Alice's  Adventures 
in  Wonderland  was  really  Rev.  Charles  Lutwidge 
Dodgson,  lecturer  in  mathematics  at  Oxford.  To  a 
building  committee  about  to  erect  a  new  school  building 
he  gave  some  advice  that  added  gaiety  to  the  delib- 
erations. Children  who  have  laughed  at  the  Mock 
Turtle's  description  of  his  school  life  in  the  sea,  as 
given  to  Alice,  will  recognize  the  same  humor  in  these 
suggestions  to  the  building  committee: 

"It  is  often  impossible  for  students  to  carry  on 
accurate  mathematical  calculations  in  close  contiguity 
to  one  another,  owing  to  their  mutual  interference  and 
a  tendency  to  general  conversation.  Consequently 
these  processes  require  different  rooms  in  which  ir- 
repressible conversationists,  who  are  found  to  occur 
in  every  branch  of  society,  might  be  carefully  and  per- 
manently fixed. 

"It  may  be  sufficient  for  the  present  to  enumerate 
the  following  requisites ;  others  might  be  added  as  the 
funds  permitted : 

"A.  A  very  large  room  for  calculating  greatest 
common  measure.  To  this  a  small  one  might  be  at- 
tached for  least  common  multiple ;  this,  however,  might 
be  dispensed  with. 

"B.   A  piece  of  open  ground  for  keeping  roots  and 

201 


202    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

practising  their  extraction;  it  would  be  advisable  to 
keep  square  roots  by  themselves,  as  their  corners  ate 
apt  to  damage  others. 

"C.  A  room  for  reducing  fractions  to  their  lowest 
terms.  This  should  be  provided  with  a  cellar  for  keep- 
ing the  lowest  terms,  when  found. 

"D.  A  large  room,  which  might  be  darkened  and 
fitted  up  with  a  magic  lantern  for  the  purpose  of  ex- 
hibiting circulating  decimals  in  the  act  of  circulation. 

"E.  A  narrow  strip  of  ground,  railed  off  and  care- 
fully leveled,  for  testing  practically  whether  parallel 
lines  meet  or  not ;  for  this  purpose  it  should  reach,  to 
use  the  expressive  language  of  Euclid,  'ever  so  far.'  " 


THE  GOLDEN  AGE  OF   MATHEMATICS. 

"The  eighteenth  century  was  philosophic,  the  nine- 
teenth scientific."  Mathematics — itself  "the  queen  of 
the  sciences,"  as  Gauss  phrased  it — is  the  necessary 
method  of  all  exact  investigation.  Kepler  exclaimed: 
"The  laws  of  nature  are  but  the  mathematical  thoughts 
of  God."  No  wonder,  therefore,  that  the  nineteenth 
century  surpassed  its  predecessors  in  extent  and  vari- 
ety of  mathematical  invention  and  application. 

One  reads  now  of  "the  recent  renaissance  of  mathe- 
matics." Strictly,  there  is  no  new  birth  or  awakening 
of  mathematics,  for  its  productivity  has  long  been 
continuous.  Being  the  index  of  scientific  progress,  it 
must  rise  with  the  rise  of  civilization.  That  rise  has 
been  so  rapid  of  late  that,  speaking  comparatively, 
one  may  be  justified  in  characterizing  the  present  great 
mathematical  activity  as  a  renaissance. 

"The  committee  appointed  by  the  Royal  Society  to 
report  on  a  catalogue  of  periodical  literature  esti- 
mated, in  1900,  that  more  than  1500  memoirs  on  pure 
mathematics   were  now   issued   annually."* 

Poets  put  the  golden  age  of  the  race  in  the  past. 
Prophets  have  seen  that  it  is  in  the  future.  The  recent 
marvelous  growth  of  mathematics  has  been  said  to 
place  its  golden  age  in  the  present  or  the  immediate 
future.    Professor  James  Pierpont,f  after  summing  up 

*  Ball,  Hist.,  p.  455. 

t  Address  before  the  department  of  mathematics  of  the  Inter- 

203 


204    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

the  mathematical  achievements  of  the  nineteenth  cen- 
tury, exclaimed:  "We  who  stand  on  the  threshold  of 
a  new  century  can  look  back  on  an  era  of  unparalleled 
progress.  Looking  into  the  future  an  equally  bright 
prospect  greets  our  eyes ;  on  all  sides  fruitful  fields  of 
research  invite  our  labor  and  promise  easy  and  rich 
returns.  Surely  this  is  the  golden  age  of  mathe- 
matics I" 

And  this  golden  age  must  last  as  long  as  men.  inter- 
rogate nature  or  value  precision  or  seek  truth.  "Mathe- 
matics is  preeminently  cosmopolitan  and  eternal." 

national  Congress  of  Arts  and  Science,  St.  Louis,  Sept.  20, 
1904,  on  "The  History  of  Mathematics  in  the  Nineteenth  Cen- 
tury," Bull.  Am.  Mathem.  Society,  11:3:  159. 


THE  MOVEMENT  TO  MAKE  MATHEMATICS 
TEACHING  MORE  CONCRETE. 

With  the  increased  mathematical  production  has 
come  a  movement  for  improved  teaching.  The  im- 
petus is  felt  in  many  lands.  "The  world-wide  move- 
ment in  the  teaching  of  mathematics,  in  the  midst  of 
which  we  stand,"  are  the  recent  words  of  a  leader  in 
this  department. * 

The  movement  is,  in  large  part,  for  more  concrete 
teaching — for  a  closer  correlation  between  the  mathe- 
matical subjects  themselves  and  between  the  mathe- 
matics and  the  natural  sciences,  for  extensive  use  of 
graphical  representation,  the  introduction  of  more  prob- 
lems pertaining  to  pupils'  interests  and  experiences,  a 
larger  use  of  induction  and  appeal  to  intuition  at  the 
expense  of  rigorous  proof  in  the  earlier  years,  the 
postponement  of  the  more  abstract  topics,  and  the 
constant  aim  to  show  the  useful  applications. 

Some  of  the  more  conservative  things  that  are  urged 
are  what  every  good  teacher  has  been  doing  for  years. 
On  the  other  hand,  some  of  the  more  radical  sugges- 
tions will  doubtless  prove  impractical  and  be  aban- 
doned. Still  the  movement  as  a  whole  is  healthful 
and  full  of  promise. 

Among  American  t  publications  that  are  taking  part 

*  Dr.  J.  W.  A.  Young,  assistant  professor  of  the  pedagogy 
of  mathematics  in  the  University  of  Chicago,  in  an  address 
before  the  mathematical  section  of  the  Central  Association  of 
Science  and  Mathematics  Teachers,  Nov.  30,  1906. 

205 


206    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

in  it  may  be  mentioned  the  magazine  School  Science 
and  Mathematics,  which  is  doing  much  for  the  cor- 
relation of  elementary  pure  and  applied  mathematics, 
the  Reports  of  the  various  committees,  the  Proceed- 
ings of  the  Central  Association  of  Science  and  Mathe- 
matics Teachers  and  similar  organizations,  and  Dr. 
Young's  new  book,  The  Teaching  of  Mathematics. 

The  Public  School  Journal  says,  "The  position  of 
mathematics  as  a  mental  tonic  would  be  strengthened," 
and  quotes  Fourier,  "The  deeper  study  of  nature  is  the 
most  fruitful  source  of  mathematical  study." 

The  movement  to  teach  the  calculus  through  en- 
gineering problems  and  the  like  has  attracted  wide 
attention. 

Some  of  the  applications  of  the  rudiments  of  de- 
scriptive geometry  to  drawing  (mechanical,  perspective 
etc.)  are  not  far  to  seek  in  works  on  drawing. 

The  applications  of  geometry  to  elementary  science 
have  been  given  in  outline.  It  would  be  well  if  there 
were  available  lists  of  the  common  applications  in  the 
trades.     E.  g.,  (in  the  carpenter's  trade)  : 

The  chalk  line  to  mark  a  straight  (etymologically, 
stretched)  line.  Illustrating  the  old  statement,  "The 
straight  line  is  the  shortest  distance  between  two 
points. " 

Putting  the  spirit  level  in  two  non-parallel  positions 
on  a  plane  surface  to  see  whether  the  surface  is  hori- 
zontal. "A  plane  is  determined  by  two  intersecting 
straight  lines.,, 

Etc. 

Perhaps  most  teachers  of  geometry  have  made,  or 
induced  pupils  to  make,  some  such  list ;  but  the  writer 
is  not  aware  that  any  extensive  compilation  is  in  print. 

Fairly  complete  lists  of  the  applications  of  algebra 


TO  MAKE  TEACHING  MORE  CONCRETE.  207 

to  the  natural  sciences  may  be  found  in  the  publications 
named  above. 

The  new  industrial  arithmetic  is  one  of  the  educa- 
tional features  of  our  time.  There  should  be  an  arith- 
metic with  problems  drawn  largely  from  agricultural 
life.  The  1905  catalog  of  the  Northern  Illinois  State 
Normal  School,  De  Kalb,  contains  a  valuable  classi- 
fied outline  of  child  activities  involving  and  illustrating 
number.  Dr.  Charles  A.  McMurry,  in  his  Special 
Method  in  Arithmetic,  mentions  the  need  of  "much 
more  abundant  statistical  data  than  the  arithmetics 
contain." 

If  we  could  have  these  things  as  teaching  material, 
without  the  affliction  of  a  fad  for  teaching  mathe- 
matics entirely  through  its  practical  applications,  it 
would  be  a  boon  indeed. 

While  rejoicing  in  the  movement  for  correlation  of 
mathematics  and  the  other  sciences,  these  two  points 
should  not  be  overlooked : 

1.  The  sciences  commonly  called  natural  are  not 
the  only  observational  sciences.  The  field  of  applied 
mathematics  is  as  broad  as  the  field  of  definite  knowl- 
edge or  investigation.  Some  parts  of  this  field  are 
specially  worthy  of  note  in  this  connection.  The  sta- 
tistical sciences,  the  social  sciences  treated  mathemat- 
ically, the  application  of  the  methods  of  exact  science 
to  social  measurements  such  .as  those  obtained  in  edu- 
cational psychology  and  the  study  of  population,  pub- 
lic health,  economic  problems  etc. — these  are  sciences 
aiming  at  accuracy.  They  seek  to  achieve  expression 
in  natural  law.  They  offer  some  of  the  best  oppor- 
tunities of  applied  mathematics.  The  recent  growth 
in  the  sciences  of  this  group  has  been,  if  possible,  more 
marked  than  that  of  the  physical  sciences.     Nor  are 


208    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

they  less  characteristic  of  the  spirit  of  our  time.  In- 
deed it  has  been  said  that  the  quotation  beginning  the 
preceding  section  should  be  extended  so  as  to  say, 
"The  eighteenth  century  was  philosophic,  the  nine- 
teenth scientific,  and  the  twentieth  is  to  be  sociologies' 

The  statistical  sciences  call  for  a  broad  acquaintance 
with  mathematical  lore  which  is  sometimes  regarded 
as  abstract  and  impractical  by  certain  critics  of  current 
mathematical  curricula.*  The  social  sciences  are  not 
studied  by  those  who  are  pursuing  elementary  mathe- 
matical courses.  It  is  not  proposed  that  elementary 
mathematics  should  be  correlated  with  them  instead 
of  with  the  physical  sciences,  or  in  addition  thereto. 
But  it  should  be  remembered  that  use  in  the  physical 
sciences  is  by  no  means  the  only  ultimate  aim  which 
makes  mathematics  practical. 

2.  The  beautiful  has  its  place  in  mathematics  as 
elsewhere.  The  p!*ose  of  ordinary  intercourse  and  of 
business  correspondence  might  be  held  to  be  the  most 
practical  use  to  which  language  is  put,  but  we  should 
be  poor  indeed  without  the  literature  of  imagination. 
Mathematics  too  has  its  triumphs  of  the  creative  im- 
agination, its  beautiful  theorems,  its  proofs  and  pro- 
cesses whose  perfection  of  form  has  made  them  classic. 

*  It  is  true  that  the  statistical  sciences  are  exposed  to  cari- 
cature, as  in  the  story  of  the  German  statistician  who  an- 
nounced that  he  had  tabulated  returns  from  the  marriage  rec- 
ords of  the  entire  country  for  the  year  and  had  discovered 
that  the  number  of  men  married  that  year  was  exactly  equal 
to  the  number  of  women  married  in  the  same  period  of  time! 
It  is  true  that  statisticians  have  (rarely)  computed  results 
that  might  have  been  deduced  a  priori.  It  is  true  also  that 
some  of  the  results  of  statistical  science  have  not  proved  to 
be  practical  or  yielded  material  returns.  But  these  things 
might  be  said  also  of  the  natural  sciences,  whose  inestimable 
value  is  everywhere  recognized.  The  social  sciences  mathe- 
matically developed  are  to  be  one  of  the  controlling  factors  in 
civilization. 


TO  MAKE  TEACHING  MORE  CONCRETE.  209 

He  must  be  a  "practical"  man  who  can  see  no  poetry 
in  mathematics ! 

Let  mathematics  be  correlated  with  physical  science ; 
let  it  be  concrete ;  but  let  the  movement  be  understood 
and  the  subject  taught  in  the  light  of  the  broadest 
educational  philosophy. 


THE  MATHEMATICAL  RECITATION  AS  AN 
EXERCISE   IN    PUBLIC   SPEAKING.*      * 

The  value  of  translating  from  a  foreign  language, 
in  broadening  the  vocabulary  by  compelling  the  mind 
to  move  in  unfrequented  paths  of  thought;  of  draw- 
ing, in  quickening  the  appreciation  of  exact  relations, 
proportion  and  perspective ;  of  the  natural  sciences,  in 
developing  independence  of  thought — this  is  all  famil- 
iar to  the  student  of  oratory;  often  has  he  been  told 
the  value  of  pursuing  these  studies  for  one  entering 
his  profession.  But  one  rarely  hears  of  the  mathemat- 
ical recitation  as  a  preparation  for  public  speaking. 
Yet  mathematics  shares  with  these  studies  their  ad- 
vantages, and  has  another  in  a  higher  degree  than 
either  of  them. 

Most  readers  will  agree  that  a  prime  requisite  for 
healthful  experience  in  public  speaking  is  that  the 
attention  of  speaker  and  hearers  alike  be  drawn  wholly 
away  from  the  speaker  and  concentrated  upon  his 
thought.  In  perhaps  no  other  class-room  is  this  so 
easy  as  in  the  mathematical,  where  the  close  reason- 
ing, the  rigorous  demonstration,  the  tracing  of  neces- 
sary conclusions  from  given  hypotheses,  commands 
and  secures  the  entire  mental  power  of  the  student 
who  is  explaining,  and  of  his  classmates.  In  what 
other  circumstances  do  students   feel  so  instinctively 

*  Article  by  the  writer  in  New  York  Education,  now  Amer- 
ican Education,  for  January,  1899 


AN  EXERCISE  IN   PUBLIC  SPEAKING.  211 

that  manner  counts  for  so  little  and  mind  for  so  much  ? 
In  what  other  circumstances,  therefore,  is  a  simple, 
unaffected,  easy,  graceful  manner  so  naturally  and  so 
healthfully  cultivated?  Mannerisms  that  are  mere 
affectation  or  the  result  of-  bad  literary  habits  recede 
to  the  background  and  finally  disappear,  while  those 
peculiarities  that  are  the  expression  of  personality  and 
are  inseparable  from  its  activity  continually  develop, 
where  the  student  frequently  presents,  to  an  audience 
of  his  intellectual  peers,  a  connected  train  of  reason- 
ing. 

How  interesting  is  a  recitation  from  this  point  of 
view !  I  do  not  recall  more  than  two  pupils  reciting 
mathematics  in  an  affected  manner.  In  both  cases  this 
passed  away.  One  of  these,  a  lady  who  was  previously 
acquainted  with  the  work  done  during  the  early  part 
of  the  term,  lost  her  mannerisms  when  the  class  took 
up  a  subject  that  was  advance  work  to  her,  and  that 
called  out  her  higher  powers. 

The  continual  use  of  diagrams  to  make  the  meaning 
clear  stimulates  the  student's  power  of  illustration. 

The  effect  of  mathematical  study  on  the  orator  in  his 
ways  of  thinking  is  apparent — the  cultivation  of  clear 
and  vigorous  deduction  from  known  facts. 

One  could  almost  wish  that  our  institutions  for  the 
teaching  of  the  science  and  the  art  of  public  speaking 
would  put  over  their  doors  the  motto  that  Plato  had 
over  the  entrance  to  his  school  of  philosophy:  "Let 
no  one  who  is  unacquainted  with  geometry  enter  here." 


THE  NATURE  OF  MATHEMATICAL  REA- 
SONING* 

Why  is  mathematics  "the  exact  science"?  Because 
of  its  self-imposed  limitations.  Mathematics  concerns 
itself,  not  with  any  problem  of  the  nature  of  things 
in  themselves,  but  with  the  simpler  problems  of  the 
relations  between  things.  Starting  from  certain  defi- 
nite assumptions,  the  mathematician  seeks  only  to 
arrive  by  legitimate  processes  at  conclusions  that  are 
surely  right  if  the  data  are  right ;  as  in  geometry. 
So  the  arithmetician  is  concerned  only  that  the  result 
of  his  computation  shall  be  correct  assuming  the  data 
to  be  correct;  though  if  he  is  also  a  teacher,  he  is  in 
that  capacity  concerned  that  the  data  of  the  problems 
set  for  his  pupils  shall  correspond  to  actual  commercial, 
industrial  or  scientific  conditions  of  the  present  day. 

Mathematics  is  usually  occupied  with  the  considera- 
tion of  only  one  or  a  few  of  the  phases  of  a  situation. 
Of  the  many  conditions  involved,  only  a  few  of  the 
most  important  and  the  most  available  are  considered. 
All  other  variables  are  treated  as  constants.  Take  for 
illustration  the  "cistern  problem, "  which  as  it  occurs 
in  the  writings  of  Heron  of  Alexandria  (c.  2d  cent. 
B.  C.)  must  be  deemed  very  respectable  on  the  score 
of  age :  given  the  time  in  which  each  pipe  can  fill  a 

vanced  section  of  teachers  institutes.  For  a  treatment  of  old 
and  new  definitions  of  mathematics,  the  reader  is  referred  to 
Prof.  Maxime  Bocher's  "The  Fundamental  Conceptions  and 
Methods  of  Mathematics,"  Bull.  Am.  Math.  Soc,  11:3:115-135. 


NATURE  OF  MATHEMATICAL  REASONING.  213 

cistern  separately,  required  the  time  in  which  they 
will  fill  it  together.  This  assumes  the  flow  to  be  con- 
stant. Other  statements  of  the  problem,  in  which 
one  pipe  fills  while  another  empties,  presuppose  the 
outflow  also  to  be  constant  whether  the  cistern  is  full 
or  nearly  empty ;  or  at  least  the  rate  of  outflow  is 
taken  as  an  average  rate  and  treated  as  a  constant. 
Or  the  "days-work  problem"  (which  is  only  the  cistern 
problem  disguised)  :  given  the  time  in  which  each  man 
can  do  a  piece  of  work  separately,  required  the  time 
in  which  they  will  do  it  together.  .This  assumes  that 
the  men  work  at  the  same  rate  whether  alone  or  to- 
gether. Some  persons  who  have  employed  labor  know 
how  violent  an  assumption  this  is,  and  are  prepared  to 
defend  the  position  of  the  thoughtless  school  boy  who 
says,  "If  A  can  do  a  piece  of  work  in  5  days  which 
B  can  do  in  3  days,  it  will  take  them  8  days  working 
together,"  as  against  the  answer  1%  days,  which  is 
deemed  orthodox  among  arithmeticians.  Or,  to  move 
up  to  the  differential  calculus  for  an  illustration :  "The 
differentials  of  variables  which  change  non-uniformly 
are  what  would  be  their  corresponding  increments  if 
at  the  corresponding  values  considered  the  change  of 
each  became  and  continued  uniform  with  respect  to  the 
same  variable."* 

Mathematics  resembles  fine  art  in  that  each  abstracts 
some  one  pertinent  thing,  or  some  few  things,  from  the 
mass  of  things  and  concentrates  attention  on  the  ele- 
ment selected.  The  landscape  painter  gives  us,  not 
every  blade  of  grass,  but  only  those  elements  that 
serve  to  bring  out  the  meaning  of  the  scene.  With 
mathematics  also  as  with  fine  art,  this  may  result  in 
a  more  valuable  product  then  any  that  could  be  ob- 
tained  by  taking   into   account   every   element.     The 

*  Taylor's  Calculus,  p.  8. 


214    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

portrait  painted  by  the  artist  does  not  exactly  repro- 
duce the  subject  as  he  was  at  any  one  moment  of  his 
life,  yet  it  may  be  a  truer  representation  of  the  man 
than  one  or  all  of  his  photographs.  So  it  is  with  one 
of  Shakespeare's  historical  dramas  and  the  annals 
which  were  its  "source/'  "The  truest  things  are  things 
that  never  happened."  , 

Mathematics  is  a  science  of  the  ideal.  The  magni- 
tudes of  geometry  exist  only  as  mental  creations,  a 
chalk  mark  being  but  a  physical  aid  to  the  mind  in 
holding  the  conception  of  a  geometric  line. 

The  concrete  is  of  necessity  complex;  only  the  ab- 
stract can  be  simple.  This  is  why  mathematics  is  the 
simplest  of  all  studies — simplest  in  proportion  to  the 
mastery  attained.  The  same  standard  of  mastery  being 
applied,  physics  is  much  simpler  than  biology:  it  is 
more  mathematical.  As  we  rise  in  the  scale  mathemat- 
ically, relations  become  simple,  until  in  astronomy  we 
find  the  nearest  approach  to  conformity  by  physical 
nature  to  a  single  mathematical  law,  and  we  see  a 
meaning  in  Plato's  dictum,  "God  geometrizes  con- 
tinually." 

Mathematics  is  thinking  God's  thought  after  him. 
When  anything  is  understood,  it  is  found  to  be  suscep- 
tible of  mathematical  statement.  The  vocabulary  of 
mathematics  "is  the  ultimate  vocabulary  of  the  material 
universe."  The  planets  had  for  many  centuries  been 
recognized  as  "wanderers"  among  the  heavenly  bod- 
ies ;  much  had  come  to  be  known  about  their  move- 
ments ;  Tycho  Brahe  had  made  a  series  of  careful 
observations  of  Mars ;  Kepler  stated  the  law :  Every 
planet  moves  in  an  elliptical  orbit  with  the  sun  at  one 
focus,  and  the  radius  vector  generates  equal  areas  in 
equal  times.    When  the  motion  was  understood,  it  was 


NATURE  OF  MATHEMATICAL  REASONING.  215 

expressed  in  the  language  of  mathematics.  Gravitation 
waited  long  for  a  Newton  to  state  its  law.  When  the 
statement  came,  it  was  in  terms  of  "the  ultimate  vocab- 
ulary" :  Every  particle  of  matter  in  the  universe  attracts 
every  other  particle  with  a  force  varying  directly  as  the 
masses,  and  inversely  as  the  square  of  the  distances. 
When  any  other  science — say  psychology — becomes  as 
definite  in  its  results,  those  results  will  be  stated  in  as 
mathematical  language.  After  many  experiments  to 
determine  the  measure  of  the  increase  of  successive 
sensations  of  the  same  kind  when  the  stimulus  in- 
creases, and  after  tireless  effort  in  the  application  of  the 
"just  perceptible  increment"  as  a  unit,  Prof.  G.  T. 
Fechner  of  Leipsic  announced  in  1860,  in  his  Psycho- 
physik,  that  the  sensation  varies  as  the  logarithm  of 
the  stimulus.  4  Fechner's  law  has  not  been  established 
by  subsequent  investigations ;  but  it  was  the  expression 
of  definiteness  in  thinking,  whether  that  thinking  was 
correct  or  not,  and  it  illustrates  mathematics  as  the 
language  of  precision. 

Mathematics,  the  science  of  the  ideal,  becomes  the 
means  of  investigating,  understanding  and  making 
known  the  world  of  the  real.  The  complex  is  ex- 
pressed in  terms  of  the  simple.  From  one  point  of  view 
mathematics  may  be  defined  as  the  science  of  successive 
substitutions  of  simpler  concepts  for  more  complex — 
a  problem  in  arithmetic  or  algebra  shown  to  depend 
on  previous  problems  and  to  require  only  the  funda- 
mental operations,  the  theorems  of  geometry  shown 
to  depend  on  the  definitions  and  axioms,  the  unknown 
parts  of  a  triangle  computed  from  the  known,  the 
simplifications  and  far-reaching  generalizations  of  the 
calculus,  etc.  It  is  true  that  we  often  have  successive 
substitutions    of   simpler   concepts    in    other    sciences 


'2l6    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

(e.  g.,  the  reduction  of  the  forms  of  logical  reasoning 
to  type  forms;  the  simplifications  culminating  in  the 
formulas  of  chemistry;  etc.)  but  we  naturally  apply 
the  adjective  mathematical  to  those  phases  of  any  sci- 
ence in  which  this  method  predominates.  In  this  view 
also  it  is  seen  why  mathematical  rigor  of  demonstration 
is  itself  an  advancing  standard.  "Archimedean  proof" 
was  to  the  Greeks  a  synonym  for  unquestionable  dem- 
onstration. 

If  a  relation  between  variables  is  stated  in  mathe- 
matical symbols,  the  statement  is  a  formula.  A  formula 
translated  into  words  becomes  a  principle  if  the  indic- 
ative mode  is  used,  a  rule  if  the  imperative  mode. 

Mathematics  is  "ultimate"  in  the  generality  of  its 
reasoning.  By  the  aid  of  symbols  it  transcends  ex- 
perience and  the  imaging  power  of  the  mind.  It  de- 
termines, for  example,  the  number  of  diagonals  of  a 
polygon  of  1000  sides  to  be  498500  by  substitution  in 
the  easily  deduced  formula  n(n-3)/2,  although  one 
never  has  occasion  to  draw  a  representation  of  a 
1000-gon  and  could  not  make  a  distinct  mental  picture 
of  its  498500  diagonals. 

If  there  are  other  inhabited  planets,  doubtless  "these 
all  differ  from  one  another  in  language,  customs  and 
laws."  But  one  can  not  imagine  a  world  in  which  ?r 
is  not  equal  to  3.14159+,  or  e  not  equal  to  2.71828+, 
though  all  the  symbols  for  number  might  easily  be 
very  different. 

In  recent  years  a  few  "astronomers,"  with  an  enter- 
prise that  would  reflect  credit  on  an  advertising  bureau, 
have  discussed  in  the  newspapers  plans  for  communi- 
cating with  the  inhabitants  of  Mars.  What  symbols 
could  be  used  for  such  communication?  Obviously 
those  which  must  be  common  to  rational  beings  every- 


NATURE  OF  MATHEMATICAL  REASONING.  2.1J 

equilateral  triangle  many  kilometers  on  a  side  and 
where.  Accordingly  it  was  proposed  to  lay  out  an 
illuminate  it  with  powerful  arc  lights.  If  our  Martian 
neighbors  should  reply  with  a  triangle,  we  could  then 
test  them  on  other  polygons.  Apparently  the  courte- 
sies exchanged  would  for  some  time  have  to  be  con- 
fined to  the  amenities  of  geometry. 

Civilization  is  humanity's  response  to  the  first — not 
the  last,  or  by  any  means  the  greatest — command  of 
its  Maker,  "Subdue  the  earth  and  have  dominion  over 
it."  And  the  aim  of  applied  mathematics  is  "the 
mastery  of  the  world  quantitatively."  "Science  is  only 
quantitative  knowledge."  Hence  mathematics  is  an 
index  of  the  advance  of  civilization. 

The  applications  of  mathematics  have  furnished  the 
chief  incentive  to  the  investigation  of  pure  mathemat- 
ics and  the  best  illustrations  in  the  teaching  of  it ;  yet 
the  mathematician  must  keep  the  abstract  science  in 
advance  of  the  need  for  its  application,  and  must  even 
push  his  inquiry  in  directions  that  offer  no  prospect 
of  any  practical  application,  both  from  the  point  of 
view  of  truth  for  truth's  sake  and  from  a  truly  far- 
sighted  utilitarian  viewpoint  as  well.  Whewell  said, 
"If  the  Greeks  had  not  cultivated  conic  sections,  Kep- 
ler could  not  have  superseded  Ptolemy."  Behind  the 
artisan  is  a  chemist,  "behind  the  chemist  a  physicist, 
behind  the  physicist  a  mathematician."  It  was  Michael 
Faraday  who  said,  "There  is  nothing  so  prolific  in 
utilities  as  abstractions." 


ALICE  IN  THE  WONDERLAND  OF  MATHE- 
MATICS. 

Years  after  Alice  had  her  "Adventures  in  Wonder- 
land" and  "Through  the  Looking-glass,"  described 
by  "Lewis  Carroll/'  she  went  to  college.  She  was  a 
young  woman  of  strong  religious  convictions.  As  she 
studied  science  and  philosophy,  she  was  often  per- 
plexed to  reduce  her  conclusions  in  different  lines  to 
a  system,  or  at  least  to  find  some  analogy  which  would 
make  the  coexistence  of  the  fundamental  conceptions 
of  faith  and  of  science  more  thinkable.  These  questions 
have  puzzled  many  a  more  learned  mind  than  hers, 
but  never  one  more  earnest. 

Alice  developed  a  fondness  for  mathematics  and 
elected  courses  in  it.  The  professor  in  that  depart- 
ment had  lectured  on  ^-dimensional  space,  and  Alice 
had  read  E.  A.  Abbott's  charming  little  book,  Flatland; 
a  Romance  of  Many  Dimensions,  by  a  Square,  which 
had  been  recommended  to  her  by  an  instructor. 

The  big  daisy-chain  which  was  to  be  a  feature  of  the 
approaching  class-day  exercises  was  a  frequent  topic 
of  conversation  among  the  students.  It  was  uppermost 
in  her  mind  one  warm  day  as  she  went  to  her  room 
after  a  hearty  luncheon  and  settled  down  in  an  easy 
chair  to  rest  and  think. 

"Why !"  she  said,  half  aloud,  "I  was  about  to  make 
a  daisy-chain  that  hot  day  when  I  fell  asleep  on  the 
bank  of  the  brook  and  went  to  Wonderland — so  long 

218 


ALICE    IN   WONDERLAND.  219 

ago.  That  was  when  I  was  a  little  girl.  Wouldn't  it 
be  fun  to  have  such  a  dream  now?  If  I  were  a  child 
again,  I'd  curl  up  in  this  big  chair  and  go  to  sleep 
this  minute.     'Let's  pretend.'  " 

So  saying,  and  with  the  magic  of  this  favorite  phrase 
upon  her,  she  fell  into  a  pleasant  revery.  Present  sur- 
roundings faded  out  of  consciousness,  and  Alice  was 
in  Wonderland. 

"What  a  long  daisy-chain  this  is!"  thought  Alice. 
"I  wonder  if  I'll  ever  come  to  the  end  of  it.  Maybe 
it  hasn't  any  end.  Circles  haven't  ends,  you  know. 
Perhaps  it's  like  finding  the  end  of  a  rainbow.  Maybe 
I'm  going  off  along  one  of  the  infinite  branches  of  a 
curve." 

Just  then  she  saw  an  arbor-covered  path  leading  off 
to  one  side.  She  turned  into  it;  and  it  led  her  into 
a  room — a  throne-room,  for  there  a  fairy  or  goddess 
sat  in  state.  Alice  thought  this  being  must  be  one 
of  the  divinities  of  classical  mythology,  but  did  not 
know  which  one.  Approaching  the  throne  she  bowed 
very  low  and  simply  said,  " Goddess" ;  whereat  that 
personage  turned  graciously  and  said,  "Welcome, 
Alice."  It  did  not  seem  strange  to  Alice  that  such  a 
being  should  know  her  name. 

"Would  you  like  to  go  through  Wonderland?" 

"Oh !  yes,"  answered  Alice  eagerly. 

"You  should  go  with  an  attendant.  I  will  send  the 
court  jester,  who  will  act  as  guide,"  said  the  fairy, 
at  the  same  time  waving  a  wand. 

Immediately  there  appeared — Alice  could  not  tell 
how — a  courtier  dressed  in  the  fashion  of  the  courts  of 
the  old  English  kings.  He  dropped  on  one  knee  before 
the  fairy ;  then,  rising  quickly,  bowed  to  Alice,  ad- 
dressing her  as,  "Your  Majesty." 


220    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

It  seemed  pleasant  to  be  treated  with  such  deference, 
but  she  promptly  answered,  "You  mistake ;  I  am  only 
Miss  — " 

Here  the  fairy  interrupted:  "Call  her  'Alice.'  The 
name  means  'princess/  " 

"And  you  may  call  me  'Phool.'  "  said  the  courtier ; 
"only  you  will  please  spell  it  with  a  ph" 

"How  can  I  spell  it  when  I  am  only  speaking  it?" 
she  asked. 

''Think  the  ph" 

"Very  well,"  answered  Alice  rather  doubtfully,  "but 
who  ever  heard  of  spelling  'fool'  with  ph?" 

Then  he  smiled  broadly  as  he  replied:  "I  am  an 
anti-spelling-reformer.  I  desire  to  preserve  the  ph 
in  words  in  place  of  f  so  that  one  may  recognize  their 
foreign  origin  and  derivation." 

"Y-e-s,"  said  Alice,  "but  what  does  phool  come 
from?" 

Again  the  fairy  interrupted.  Though  always  gra- 
cious, she  seemed  to  prefer  brevity  and  directness. 
"You  will  need  the  magic  wand." 

So  saying,  she  handed  it  to  the  jester.  The  moment 
he  had  the  wand,  the  fairy  vanished.  And  the  girl  and 
the  courtier  were  alone  in  the  wonderful  world,  and 
they  were  not  strangers.  They  were  calling  each  other 
"Alice"  and  "Phool."     And  he  held  the  magic  wand. 

One  flourish  of  that  wand,  and  they  seemed  to  be 
in  a  wholly  different  country.  There  were  many 
beings,  having  length,  but  no  breadth  or  thickness ; 
or,  rather,  they  were  very  thin  in  these  two  dimen- 
sions, and  uniformly  so.  They  were  moving  only  in 
one  line. 

"Oh !  I  know !"  exclaimed  Alice,  "This  is  Lineland. 
I  read  about  it." 


ALICE    IN   WONDERLAND.  221 

"Yes,"  said  Phool ;  "if  you  hadn't  read  about  it  or 
thought  about  it,  I  couldn't  have  shown  it  to  you." 

Alice  looked  questioningly  at  the  wand  in  his  hand. 

"It  has  marvelous  power,  indeed,"  he  said.  "To 
show  you  in  this  way  what  you  have  thought  about, 
that  is  magic ;  to  show  you  what  you  had  never  thought 
of,  would  be—" 

Alice  could  not  catch  the  last  word.  A  little  twitch 
of  the  wand  set  them  down  at  a  different  point  in  the 
line,  where  they  could  get  a  better  view  of  lineland. 
Alice  thrust  her  hand  across  the  line  in  front  of  one  of 
the  inhabitants.  He  stopped  short.  She  withdrew  it. 
He  was  amazed  at  the  apparition:  a  body  (or  point) 
had  suddenly  appeared  in  his  world  and  as  suddenly 
vanished.  Alice  was  interested  to  see  how  a  line- 
lander  could  be  imprisoned  between  two  points. 

"He  never  thinks  to  go  around  one  of  the  obstacles," 
she  said. 

"The  line  is  his  world,"  said  Phool.  "One  never 
thinks  of  going  out  of  the  world  to  get  around  an 
obstacle." 

"If  I  could  communicate  with  him,  could  I  teach 
him  about  a  second  dimension?" 

"He  has  no  apperceiving  mass,"  said  Phool  lacon- 
ically. 

"Very  good,"  said  Alice,  laughing;  "surely  he  has 
no  mass.  Then  he  can  get  out  of  his  narrow  world 
only  by  accident?" 

"Accident!"  repeated  Phool,  affecting  surprise,  "I 
thought  you  were  a  philosopher." 

"No,"  replied  Alice,  "I  am  only  a  college  girl." 

"But,"  said  Phool,  "you  are  a  lover  of  wisdom. 
Isn't  that  what  'philosopher'  means?  You  see  I'm  a 
stickler  for  etymologies." 


222    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

"All  right/'  said  Alice,  "I  am  a  philosopher  then. 
But  tell  me  how  that  being  can  ever  appreciate  space 
outside  of  his  world." 

"He  might  evolve  a  few  dimensions." 

Alice  stood  puzzled  for  a  minute,  though  she  knew 
that  Phool  was  jesting.  Then  a  serious  look  came 
into  his  face,  and  he  continued: 

"One-dimensional  beings  can  learn  of  another  dimen- 
sion only  by  the  act  of  some  being  from  without  their 
world.    But  let  us  see  something  of  a  broader  world." 

So  saying,  he  waved  the  wand,  and  they  were  in  a 
country  where  the  inhabitants  had  length  and  breadth, 
but  no  appreciable  thickness. 

Alice  was  delighted.  "This  is  Flatland,"  she  cried 
out.  Then  after  a  minute  she  said,  "I  thought  the 
Flatlanders  were  regular  geometric  figures." 

Phool  laughed  at  this  with  so  much  enjoyment  that 
Alice  laughed  too,  though  she  saw  nothing  very  funny 
about  it. 

Phool  explained :  "You  are  thinking  of  the  Flatland 
where  all  lawyers  are  square,  and  where  acuteness 
is  a  characteristic  of  the  lower  classes  while  obtuseness 
is  a  mark  of  nobility.  That  would,  indeed,  be  very 
flat;  but  we  spell  that  with  a  capital  F.  This  is  flat- 
land  with  a  small  f" 

Alice  fell  to  studying  the  life  of  the  two-dimension 
people  and  thinking  how  the  world  must  seem  to  them. 
She  reasoned  that  polygons,  circles  and  all  other  plane 
figures  are  always  seen  by  them  as  line-segments ;  that 
they  can  not  see  an  angle,  but  can  infer  it ;  that  they 
may  be  imprisoned  within  a  quadrilateral  or  any  other 
plane  figure  if  it  has  a  closed  perimeter  which  they 
may  not  cross ;  and  that  if  a  three-dimensional  being 
were  to  cross  their  world   (surface)   they  could  ap- 


ALICE    IN   WONDERLAND.  223 

preciate  only  the  section  of  him  made  by  that  surface, 
so  that  he  would  appear  to  them  to  be  two-dimensional 
but  possessing  miraculous  powers  of  motion. 

Alice  was  pleased,  but  curious  to  see  more.  "Let's 
see  other  dimensional  worlds,"  she  said. 

"Well,  the  three-dimensional  world,  you're  in  all 
the  time,"  said  Phool,  at  the  same  time  moving  the 
wand  a  little  and  changing  the  scene,  "and  now  if  you 
will  show  me  how  to  wave  this  wand  around  through 
a  fourth  dimension,  we'll  be  in  that  world  straight- 
way." 

"Oh !  I  can't,"  said  Alice. 

"Neither  can  I,"  said  he. 

"Can  anybody?" 

"They  say  that  in  four-dimensional  space  one'  can 
see  the  inside  of  a  closed  box  by  looking  into  it  from 
a  fourth  dimension  just  as  you  could  see  the  inside  of 
a  rectangle  in  flatland  by  looking  down  into  it  from 
above ;  that  a  knot  can  not  be  tied  in  that  space ;  and 
that  a  being  coming  to  our  world  from  such  a  world 
would  seem  to  us  three-dimensional,  as  all  we  could 
see  of  him  would  be  a  section  made  by  our  space,  and 
that  section  would  be  what  we  call  a  solid.  He  would 
appear  to  us — let  us  say — as  human.  And  he  would 
be  not  less  human  than  we,  nor  less  real,  but  more  so ; 
if  'real'  has  degrees  of  comparison.  The  flatlander 
who  crosses  the  linelander's  world  (line)  appears  to 
the  native  to  be  like  the  one-dimensional  beings,  but 
possessed  of  miraculous  powers.  So  also  the  solid  in 
flatland:  the  cross-section  of  him  is  all  that  a  flat- 
lander  is,  and  that  is  only  a  section,  only  a  phase  of 
his  real  self.  The  ability  of  a  being  of  more  than 
three  dimensions  to  appear  and  disappear,  as  to  enter 
or  leave  a   room   when  all   doors   were   shut,   might 


224    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

make  him  seem  to  us  like  a  ghost,  but  he  would  be 
more  real  and  substantial  than  we  are." 

He  paused,  and  Alice  took  occasion  to  remark : 

"That  is  all  obtained  by  reason ;  I  want  to  see  a 
four-dimensional  world." 

Then,  fearing  that  it  might  not  seem  courteous  to 
her  guide  to  appear  disappointed,  she  added: 

"But  I  ought  to  have  known  that  the  wand  couldn't 
show  us  anything  we  might  wish  to  see ;  for  then  there 
would  be  no  limit  to  our  intelligence." 

"Would  unlimited  intelligence  mean  the  same  thing 
as  absolutely  infinite  intelligence  ?"  Phool  asked. 

"That  sounds  to  me  like  a  conundrum/'  said  Alice. 
"Is  it  a  play  on  words?" 

"There  goes  Calculus,"  said  Phool.  "Fll  ask  him. — 
Hello!  Cal." 

Alice  looked  and  saw  a  dignified  old  gentleman 
with  flowing  white  beard.  He  turned  when  his  name 
was  called. 

While  Calculus  was  approaching  them,  Phool  said 
in  a  low  tone  to  Alice:  "He'll  enjoy  having  an  eager 
pupil  like  you.    This  will  be  a  carnival  for  Calculus." 

When  that  worthy  joined  them  and  was  made  ac- 
quainted with  the  topic  of  conversation,  he  turned  to 
Alice  and  began  instruction  so  vigorously  that  Phool 
said,  by  way  of  caution: 

"Lass!  Handle  with  care." 

Alice  did  not  like  the  implication  that  a  girl  could 
not  stand  as  much  mathematics  as  any  one.  But  then 
she  thought,  "That  is  only  a  joke,"  and  she  seemed 
vaguely  to  remember  having  heard  it  somewhere  be- 
fore. 

"If  you  mean,"  said  Calculus,  "to  ask  whether  a 
variable  that  increases  without  limit  is  the  same  thing 


ALICE    IN   WONDERLAND.  225 

as  absolute  infinity,  the  answer  is  clearly  No.  A 
variable  increasing  without  limit  is  always  nearer  to 
zero  than  to  absolute  infinity .  For  simplicity  of  illus- 
tration, compare  it  with  the  variable  of  uniform  change, 
time,  and  suppose  the  variable  we  are  considering 
doubles  every  second.  Then,  no  matter  how  long  it 
may  have  been  increasing  at  this  rate,  it  is  still  nearer 
zero  than  infinity. " 

"Please  explain/'  said  Alice. 

"Well,"  continued  Calculus,  "consider  its  value  at 
any  moment.  It  is  only  half  what  it  will  be  one  second 
hence,  and  only  quarter  what  it  will  be  two  seconds 
hence,  when  it  will  still  be  increasing.  Therefore  it 
is  nozv  much  nearer  zero  than  infinity.  But  what  is 
true  of  its  value  at  the  moment  under  consideration  is 
true  of  any,  and  therefore  of  every,  moment.  An  in- 
finite is  always  nearer  to  zero  than  to  infinity." 

"Is  that  the  reason,"  asked  Alice,  "why  one  must 
say  'increases  without  limit'  instead  of  'approaches 
infinity  as  a  limit'?" 

"Certainly,"  said  Calculus ;  "a  variable  can  not  ap- 
proach infinity  as  a  limit.  Students  often  have  to  be 
reminded  of  this." 

Alice  had  an  uncomfortable  feeling  that  the  con- 
versation was  growing  too  personal,  and  gladly  turned 
it  into  more  speculative  channels  by  remarking: 

"I  see  that  one  could  increase  in  wisdom  forever, 
though  that  seems  miraculous." 

"What  do  you  mean  by  miraculous?"  asked  Phool. 

"Wiry — "  began  Alice,  and  hesitated. 

"People  who  begin  an  answer  with  'Why'  are  rarely 
able  to  give  an  answer,"  said  Phool. 

"I  fear  I  shall  not  be  able,"  said  Alice.  "An  ety- 
mologist" (this  with  a  sly  look  at  Phool)  "might  say 


226    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

it  means  'wonderful' ;  and  that  is  what  I  meant  when 
speaking  about  infinites.  But  usually  one  would  call 
that  miraculous  which  is  an  exception  to  natural  law." 

"We  must  take  the  young  lady  over  to  see  the  curve 
tracing,"  said  Calculus  to  Phool. 

"Yes,  indeed!"  he  replied.  Then,  turning  to  Alice, 
"Do  you  enjoy  fireworks?" 

"Yes,  thank  you,"  said  Alice,  "but  I  can't  stav  till 
dark." 

"No?"  said  Phool,  with  an  interrogation.  "Well, 
we'll  have  them  very  soon." 

"Fireworks  in  daytime?"  she  asked. 

But  at  that  moment  Phool  made  a  flourish  with  the 
wand,  and  it  was  night — a  clear  night  with  no  moon 
or  star.  It  seemed  so  natural  for  the  magic  wand  to 
accomplish  things  that  Alice  was  not  very  much  sur- 
prised at  even  this  transformation.     She  asked : 

"Did  you  say  you  were  to  show  me  curve  tracing  ?" 

"Yes,"  said  Phool.  "Perhaps  you  don't  attend  the 
races,  but  you  may  enjoy  seeing  the  traces/' 

During  this  conversation  the  three  had  been  walk- 
ing, and  they  now  came  to  a  place  where  there  was 
what  appeared  to  be  an  enormous  electric  switchboard. 
A  beautiful  young  woman  was  in  charge. 

As  they  approached,  Calculus  said  to  Alice,  "That 
is  Ana  Lytic.  You  are  acquainted  with  her,  I  pre- 
sume." 

"The  name  sounds  familiar,"  said  Alice,  "but  I  don't 
remember  to  have  ever  seen  her.  I  should  like  to 
meet  her." 

On  being  presented,  Alice  greeted  her  new  acquaint- 
ance as  'Miss  Lytic' ;  but  that  person  said,  in  a  very 
gracious  manner: 

"Nobody  ever  addresses  me  in  that  way.     I  am  al- 


ALICE    IN   WONDERLAND.  227 

ways  called  'Ana  Lytic/  except  by  college  students. 
They  usually  call  me  'Ana  Lyt.'  I  presume  they 
shorten  my  name  thus  because  they  know  me  so  well.', 

In  spite  of  the  speaker's  winning  manner,  the  last 
clause  made  Alice  somewhat  self  -  conscious.  Her 
cheeks  felt  very  warm.  She  was  relieved  when,  at  that 
moment,  Calculus  said : 

"This  young  lady  would  like  to  see  some  of  your 
work/' 

"Some  pyrotechnic  curve  tracing,"  interrupted  the 
talkative  Phool. 

Calculus  continued:  "Please  let  us  have  an  algebraic 
curve  with  a  conjugate  point." 

Ana  Lytic  touched  a  button,  and  across  the  world 
of  darkness  (as  it  seemed  to  Alice)  there  flashed  a 
sheet  of  light,  dividing  space  by  a  luminous  plane. 
It  quickly  faded,  but  left  two  rays  of  light  perpen- 
dicular to  each  other,  faint  but  apparently  permanent. 

"These  are  the  axes  of  coordinates,"  explained  Ana 
Lytic. 

Then  she  pressed  another  button,  and  Alice  saw 
what  looked  like  a  meteor.  She  watched  it  come  from 
a  great  distance,  cross  the  ray  of  light  that  had  been 
called  one  of  the  axes,  and  go  off  on  the*  other  side 
as  rapidly  as  it  had  come,  always  moving  in  the  plane 
indicated  by  the  vanished  sheet  of  light.  She  thought 
of  a  comet ;  but  instead  of  having  merely  a  luminous 
tail,  it  left  in  its  wake  a  permanent  path  of  light.  Ana 
Lytic  had  come  close  to  Alice,  and  the  two  girls  stood 
looking  at  the  brilliant  curve  that  stretched  away 
across  the  darkness  as  far  as  the  eye  could  reach. 

"Isn't  it  beautiful !"  exclaimed  Alice. 

Any  attempt  to  represent  on  paper  what  she  saw 


228    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

must  be  poor  and  inadequate.     Figure  58  is  such  an 
attempt. 

Suddenly  she  exclaimed:  "What  is  that  point  of 
light  ?"  indicating  by  gesture  a .  bright  ooint  situated 
as  shown  in  the  figure  by  P. 


a- 


0 


r 


Fig.  58. 


''That  is  a  point  of  the  curve,"  said  Ana  Lytic. 

"But  it  is  away  from  all  the  rest  of  it,"  objected 
Alice. 

Going  over  to  her  apparatus  and  taking  something — 
Alice  could  not  see  what — Ana  Lytic  began  to  write 
on   what,   in  the  darkness,  might   surely  be  called  a 


ALICE    IN   WONDERLAND.  229 

blackboard.  The  characters  were  of  the  usual  size 
of  writing  on  school  boards,  but  they  were  characters 
of  light  and  could  be  plainly  read  in  the  night.  This 
is  what  sne  wrote: 

r>=(.r-2)2(.r-3). 

Stepping  back,  she  said:  "That  is  the  equation  of 
the  curve." 

Alice  expressed  her  admiration  at  seeing  the  equa- 
tion before  her  and  its  graph  stretching  across  the 
world  in  a  line  of  light. 

"I  never  imagined  coordinate  geometry  could  be  so 
beautiful,"  she  said. 

"This  is  throwing  light  on  the  subject  for  you," 
said  Phool. 

"The  point  about  which  you  asked,"  said  Ana  Lytic 
to  Alice,  "is  the  point  (2,0).  You  see  that  it  satisfies 
the  equation.     It  is  a  point  of  the  graph." 

Alice  now  noticed  that  units  of  length  were  marked 
off  on  the  dimly  seen  axes  by  slightly  more  brilliant 
points  of  light.  Thus  she  easily  read  the  coordinates 
of  the  point. 

"Yes,"  she  said,  "I  see  that ;  but  it  seems  strange 
that  it  should  be  off  away  from  the  rest." 

"Yes,"  said  Calculus,  who  had  been  listening  all  the 
time.  "One  expects  the  curve  to  be  continuous.  Con- 
tinuity is  the  message  of  modern  scientific  thought. 
This  point  seems  to  break  that  law — to  be  'miraculous,' 
as  you  defined  the  term  a  few  minutes  ago.  If  all 
observed  instances  but  one  have  some  visible  con- 
nection, we  are  inclined  to  call  that  one  miraculous 
and  the  rest  natural.  As  only  that  seems  wonderful 
which  is  unusual,  the  miraculous  in  mathematics  would 
be  only  an  isolated  case." 

"I  thank  you,"  said  Alice  warmly.     "That  is  the 


23O    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

way  I  should  like  to  have  been  able  to  say  it.  An 
isolated  case  is  perplexing  to  me.  I  like  to  think  that 
there  is  a  universal  reign  of  law." 

"Evidently"  said  Phool,  "here  is  an  exception.  It 
is  obvious  that  there  are  several  alternatives,  such  as, 
for  example,  that  the  point  is  not  on  the  graph,  that 
the  graph  has  an  isolated  point,  and  so  forth" 

Calculus,  Ana  Lytic  and  Phool  all  laughed  at  this. 
To  Alice's  inquiry,  Phool  explained: 

"We  often  say  'evidently'  or  'obviously'  when  we 
can't  give  a  reason,  and  we  conclude  a  list  with  'and 
so  forth'  when  we  can't  think  of  another  item." 

Alice  felt  the  remark  might  have  been  aimed  at  her. 
Still  she  had  not  used  either  of  these  expressions  in  this 
conversation,  and  Phool  had  made  the  remark  in  a 
general  way  as  if  he  were  satirizing  the  foibles  of  the 
entire  human  race.  Moreover,  if  she  felt  inclined  to 
resent  it  as  an  impertinent  criticism  from  a  self-con- 
stituted teacher,  she  remembered  that  it  was  only  the 
jest  of  a  jester  and  treated  it  merely  as  an  interruption. 

"Tell  me  about  the  isolated  point,"  she  said  to 
Calculus. 

He  proceeded  in  a  teacher-like  way,  which  seemed 
appropriate  in  him. 

Calculus.  For  x-2m  this  equation,  y = 0.  For  any 
other  value  of  x  less  than  3,  what  would  y  be? 

Alice.    An  imaginary. 

Calculus.  And  what  is  the  geometric  representa- 
tion of  an  imaginary  number? 

Alice.  A  line  whose  length  is  given  by  the  absolute, 
or  arithmetic,  value  of  the  imaginary  and  whose  direc- 
tion is  perpendicular  to  that  which  represents  positives 
and  negatives. 

Calculus.     Good.     Then — 


ALICE    IN   WONDERLAND. 


231 


Alice  (bounding  with  delight  at  the  discovery).  Oh! 
I  see !  I  see !  There  must  be  points  of  the  graph  out- 
side of  the  plane. 

Calculus.  Yes,  there  are  imaginary  branches,  and 
perhaps  Ana  Lytic  will  be  good  enough  to  show  you 
now. 


v  Y 


X1 


<y 


Q'w 


\ 


\ 


\ 

\s 

\ 

% 


w  ,--\ 


TfcO 


I 


lc 


I 


Fig.  59- 


The  dotted  line  QPQ,  if  revolved  900  about  XX'  as  axis,  re- 
maining in  that  position  in  plane  perpendicular  to  paper,  would 
be  the  "imaginary  part"  of  the  graph. 

The  dot-and-dash  line  SRPRS  represents  the  projection 
on  the  plane  of  the  paper  of  the  two  "complex  parts."  At  P 
each  branch  is  in  the  plane  of  the  paper,  at  each  point  R  one 
branch  is  about  0.7  from  the  plane  each  side  of  the  paper,  at  S 
each  branch  is  1.5  from  the  plane,  etc. 


232    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

That  young  lady  touched  something  on  her  magic 
switchboard,  and  another  brilliant  curve  stretched 
across  the  heavens.  The  plane  determined  by  it  was 
perpendicular  to  the  plane  previously  shown.  (The 
dotted  line  in  Fig.  59  represents  in  a  prosaic  way 
what  Alice  saw.) 

"O,  I  see!"  exclaimed  Alice.  "That  point  is  not 
isolated.  It  is  the  point  in  which  this  'imaginary' 
branch,  which  is  as  real  as  any,  pierces  the  plane  of 
the  two  axes." 

"Now,"  said  Calculus,  "if  instead  of  substituting 
real  values  for  x  and  solving  the  equation  for  y,  you 
were  to  substitute  real  numbers  for  y  and  solve  for  x, 
you  would,  in  general,  obtain  for  each  value  of  y  one 
real  and  two  complex  numbers  as  the  values  of  x. 
The  curve  through  all  the  points  with  complex  ab- 
scissas is  neither  in  the  plane  of  the  axes  nor  in  a 
plane  perpendicular  to  it.     But  you  shall  see." 

(The  dot-and-dash  line  in  Fig.  59  represents  these 
branches.) 

When  Ana  Lytic  made  the  proper  connection  at  the 
switchboard,  these  branches  of  the  curve  also  stood 
out  in  lines  of  light. 

Alice  was  more  deeply  moved  than  ever.  There  was 
a  note  of  deep  satisfaction  in  her  voice  as  she  said : 

"The  point  that  troubled  me  because  of  its  isolation 
is  a  point  common  to  several  branches  of  the  curve." 

"The  supernatural  is  more  natural  than  anything 
else,"  said  Phool. 

"The  miraculous,"  thought  Alice,  "is  only  a  special 
case  of  a  higher  law.  We  fail  to  understand  things 
because  they  are  connected  with  that  which  is  out  of 
our  plane." 


ALICE    IN    WONDERLAND.  233 

She  added  aloud:  "This  I  should  call  the  miracle 
curve." 

"Yet  there  is  nothing  exceptional  about  this  curve," 
said  Calculus.  "Any  algebraic  curve  with  a  conjugate 
point  has  similar  properties." 

Then  Calculus  said  something  to  Ana  Lytic — Alice 
could  not  hear  what — and  Ana  Lytic  was  just  touching 
something  on  the  switchboard  when  there  was  a  crash 
of  thunder.  Alice  gave  a  start  and  awoke  to  find  her- 
self in  her  own  room  at  midday,  and  to  realize  that 
the  slamming  of  a  door  in  the  corridor  had  been  the 
thunder  that  terminated  her  dream. 

She  sat  up  in  the  big  chair  and,  with  the  motion 
that  had  been  characteristic  of  her  as  a  little  girl,  gave 
"that  queer  little  toss  of  her  head,  to  keep  back  the 
wandering  hair  that  zvould  always  get  into  her  eyes," 
and  said  to  herself: 

"There  aren't  any  curves  of  light  across  the  sky 
at  all !  And  worlds  of  one  or  two  dimensions  exist 
only  in  the  mind.  They  are  abstractions.  But  at  least 
they  are  thinkable.  I'm  glad  I  had  the  dream.  Imag- 
ination is  a  magic  wand. — The  future  life  will  be  a 
real  wonderland,  and — " 

Then  the  ringing  of  a  bell  reminded  her  that  it  was 
time  to  start  for  an  afternoon  lecture,  and  she  heard 
some  of  her  classmates  in  the  corridor  calling  to  her, 
"Come,  Alice." 


BIBLIOGRAPHIC  NOTES. 

Mathematical  recreations.  The  Ahmes  papyrus,  oldest  math- 
ematical work  in  existence,  has  a  problem  which  Cantor  inter- 
prets as  one  proposed  for  amusement.  At  which  Cajori  re- 
marks :*  "If  the  above  interpretations  are  correct,  it  looks  as 
if  'mathematical  recreations'  were  indulged  in  by  scholars 
forty  centuries  ago." 

The  collection  of  "Problems  for  Quickening  the  Mind"  Can- 
tor thinks  was  by  Alcuin  (735-804).  Cajori's  interesting 
commentf  is :  "It  has  been  remarked  that  the  proneness  to 
propound  jocular  questions  is  truly  Anglo-Saxon,  and  that 
Alcuin  was  particularly  noted  in  this  respect.  Of  interest  is 
the  title  which  the  collection  bears :  'Problems  for  Quickening 
the  Mind.'  Do  not  these  words  bear  testimony  to  the  fact 
that  even  in  the  darkness  of  the  Middle  Ages  the  mind- 
developing  power  of  mathematics  was  recognized?" 

Later  many  collections  of  mathematical  recreations  were 
published,  and  many  arithmetics  contained  some  of  the  recrea- 
tions. Their  popularity  is  noticeable  in  England  and  Germany 
in  the  seventeenth  and  eighteenth  centuries.J 

A  good  bibliography  of  mathematical  recreations  is  given 
by  Lucas. §  There  are  16  titles  from  the  sixteenth  century,  33 
from  the  seventeenth,  38  from,  the  eighteenth,  and  100  from 
the   nineteenth    century,   the   latest    date   being    1890.      Young 

*  Hist,  of  Elem.  Math.,  p.  24.  f  Id.,  p.  1 13-4. 

t  A  book  entitled  Rara  Arithmetica,  by  Prof.  David  Eugene  Smith, 
is  to  be  published  by  Ginn  &  Co.  the  coming  summer  or  fall  (1907). 
It  will  contain  six  or  seven  hundred  pages  and  have  three  hundred  illus- 
trations, presenting  graphically  the  most  interesting  facts  in  the  history 
of  arithmetic.  Its  author's  reputation  in  this  field  insures  the  book 
an  immediate  place  among  the  classics  of  mathematical   history. 

§  1:237-248.  Extensive  as  his  list  is,  it  is  professedly  restricted  in 
scope.  He  says.  Nous  donnons  ci-apres,  suivant  l'ordre  chronologique, 
l'indication  des  principaux  livres,  memoires,  extraits  de  correspondence, 
qui  ont  ete  publies  sur  l'Arithmetique  de  position  et  sur  la  Geometrie  de 
situation.  Nous  avons  surtout '  choisi  les  documents  qui  se  rapportent 
aux  sujets  que  nous  avons  traites  ou  que  nous  traiterons  ulterieurement. 

234 


BIBLIOGRAPHIC  NOTES.  235 

(p-  l73~4)  gives  a  list  of  20  titles,  mostly  recent,  in  no  case 
duplicating  those  of  Lucas's  list  (except  where  mentioning  a 
later  edition).  This  gives  a  total  of  over  two  hundred  titles. 
Now  turn  to  two  other  collections,  and  we  find  the  list  greatly 
extended.  Ahrens'  Mathcmatische  Unterhaltungen  (1900)  has 
a  bibliography  of  330  titles,  including  nearly  all  those  given 
by  Lucas.  Fourrey's  Curiositccs  Geometriques  (1907)  has  the 
most  recent  bibliography.  It  is  extensive  in  itself  and  mostly 
supplementary  to  the  lists  by  Lucas  and  Ahrens. 

In  all  the  vast  number  of  published  mathematical  recrea- 
tions, the  present  writer  does  not  know  of  a  book  covering 
the  subject  in  general  which  was  written  and  published  in 
America.  We  seem  to  have  taken  our  mathematics  very  seri- 
ously on  this  side  of  the  Atlantic. 

Publications  of  foregoing  sections  in  periodicals.  The  sec- 
tions of  this  book  which  have  been  printed  in  magazines  are 
as  follows.  The  month  and  year  in  each  case  are  those  of  the 
magazine,  and  the  page  is  the  page  of  this  book  at  which  the 
section  begins. 

The  Open  Court,  January  1907,  p.  218;  February,  p.  212; 
March,  p.  73,  76;  April,  p.  109;  May,  p.  143,  154,  196,  122; 
June,  p.  81,  83;  July,  p.  168,  170. 

The  Monist,  January  1907,  p.   11,  15. 

New  York  Education  (now  American  Education),  January 
1899,  p.  210. 

American    Education,   September    1906,   p.   59;    March    1907, 

p.  51. 

Some  of  the  articles  have  been  altered  slightly  since  their 
publication  in  periodical  form. 


BIBLIOGRAPHIC  INDEX. 

List  of  the  publications  mentioned  in  this  book,  with  the  pages  where 
mentioned.  The  pages  of  this  book  are  given  after  the  imprint  in  each 
entry.     These  references  are  not  included  in  the  general  index. 

A  date  in   (  )   is  the  date  of  copyright. 

In  many  cases  a  work  is  barely  mentioned.  *  indicates  either  more 
extended  use  made  of  the  book  in  this  case,  or  'direct  (though  brief) 
quotation,  or  a  figure  taken  from  the  book. 

[Abbott,  E.  A.]  Flatland;  a  Romance  of  Many  Dimensions, 
by  a  Square.     [London,  1884]  Boston,  1899.     *2i8. 

Ahrens.  Mathematische  Unterhaltnngen  und  Spiele.  Leip- 
zig, 1900.    235. 

American  Education    (monthly).     Albany,   N.   Y.     145,  *2io, 

235. 

Annali  di  Matematica.     Milan.    38. 

Argand,  J.  R.     Essai.    Geneva,  1806.    94 

Ball,  W.  W.  R.  Mathematical  Recreations  and  Essays.  Ed.  4. 
Macmillan,  London,  1905.  (A  book  both  fascinating  and 
scholarly,  attractive  to  every  one  with  any  taste  for 
mathematical  studies.)  *35,*38,  *4i,  *83,  III,  117,  *I22, 
123,  *I27,  *i4i,  *i7i,  186,  187,  *200 

Ball,  W.  W.  R.  Short  Account  of  the  History  of  Mathematics. 
Ed.  3.     Macmillan,  London,   1901.     *34,  *35,  ^37,  *i23, 

*203 

Beman  and  Smith.  New  Plane  Geometry.  Ginn  (1895,  '99). 
164 

Bledsoe,  A.  T.  Philosophy  of  Mathematics.  Lippincott,  1891 
(1867).     150. 

Brooks,  Edward.  Philosophy  of  Arithmetic.  .  .Sower,  Phila- 
delphia (1876.)  (An  admirable  popular  presentation  of 
some  of  the  elementary  theory  of  numbers ;  also  his- 
torical notes).    25,  31,  *5o,  66. 

Bruce,  W.  H.  Some  Noteworthy  Properties  of  the  Triangle 
and  its  Circles.  Heath,  1903.  (One  of  the  series  of 
Heath's  Mathematical  Monographs,  10  cents  each).  135 
236 


BIBLIOGRAPHIC  INDEX.  237 

Bulletin   of  the   American   Mathematical    Society    (monthly). 

Lancaster,  Pa.,  and  New  York  City.     *I03,  *204,  212. 
Cajori,    Florian.      History   of   Elementary   Mathematics,   with 

Hints  on  Methods  of  Teaching.    Macmillan,  1905  (1896). 

(This  suggestive  book  should  be  read  by  every  teacher.) 

*52,  *67,  *gi,  135,  148,  *i65,  193,  195,  *234. 
Cajori,  Florian.     History  of  Mathematics.     Macmillan,   1894. 

37,   148,   193. 
Cantor,  Moritz.    Vorlesungen  iiber  die  Geschichte  der  Mathe- 

matik.     3  vol.     Teubner,  Leipzig,   1880-92.     49,  67,   148., 

234. 

De  Morgan,  Augustus.     Arithmetical  Books.     68. 

De   Morgan,   Augustus.      Budget   of    Paradoxes.      Longmans, 
London,  1872.    *35,  *4i,  86,  *i26,  *i8i. 

Dietrichkeit,  O.  Siebenstellige  Logarithmen  und  Antiloga- 
rithmen.     Julius  Springer,  Berlin,  1903.     40. 

Dodgson,  C.  L.  Alice's  Adventures  in  Wonderland.  1865. 
201,  *2l8. 

Dodgson,  C.  L.  Through  the  Looking-glass  and  What  Alice 
Found  There.     1872.     *2i8. 

Encyclopaedia  Britannica.     Ed.  9.     39,  *yi,  ^176,  *i83,  186. 

Euler,  Leonhard.  Solutio  Problematis  ad  Geometriam  Situs 
Pertinentis.     St.  Petersburg,  1736.     170. 

Evans,  E.  P.  Evolutional  Ethics  and  Animal  Psychology. 
Appleton,  1898.     119. 

Fechner,  G.  T.     Psychophysik.     i860.     215. 

Fink,  Karl.  Brief  History  of  Mathematics,  tr.  by  Beman  and 
Smith.     Open  Court  Publishing  Co.,  1900.    49,  93,  148. 

Fourier.  Analyse  des  Equations  Determinees.  23. 

Fourrey,  E.  Curiositees  Geometriques.  Vuibert  et  Nony, 
Paris,  1907.     235. 

Girard,  Albert.  Invention  Nouvelle  en  l'Algebre.  Amsterdam, 
1629.     92. 

Gray,  Peter.  Tables  for  the  Formation  of  Logarithms  and 
Antilogarithms  to  24  or  any  less  Number  of  Places. 
C.  Layton,  London,  1876.    40. 

Halsted,  G.  B.  Bibliography  of  Hyperspace  and  Non-Euclid- 
ean Geometry.     1878.     104,  107. 

Harkness,  William.  Art  of  Weighing  and  Measuring.  Smith- 
sonian Report  for  j888.    ^43. 


238    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

Hooper,  W.  Rational  Recreations,  in  which  the  Principles  of 
Numbers  and  Natural  Philosophy  Are  Clearly  and 
Copiously  Elucidated.  ..  .4  vol.  London,  1774.  (Only 
the  first  166  pages  of  vol.  1  treat  of  numbers.)     26,  *27, 

.    *38 
Journal  of  the  American  Medical  Association.     Chicago.     158. 
Kempe,  A.  B.     How  to  Draw  a  Straight  Line ;  a  Lecture  on 

Linkages.     Macmillan,  London,   1877.     *I32,  ^136,  ^139. 
Klein,  F.     Famous  Problems  of  Elementary  Geometry;  tr.  by 

Beman  and  Smith.     Ginn,  1897.     I23- 
Knowledge.     187. 
Lagrange,   J.   L.     Lectures   on    Elementary   Mathematics ;    tr. 

by  T.  J.  McCormack.     Ed.  2.     Open  Court  Publishing 

Co.,  1901    (1898).     61. 
Lebesgue,  V.   A.     Table   des   Diviseurs   des   Nombres.     Gau- 

thier-Villars,  Paris.     40. 
Leonardo  Fibonacci.     Algebra  et  Almuchabala  (Liber  Abaci). 

1202.    66. 
LTntermediaire  des  Mathematiciens.  *20,  *2i,  36. 
Listing,  J.  B.    Vorstudien  zur  Topologie  (Abgedruckt  aus  den 

Gottinger  Studien).     Gottingen,  1848.     117,  170,   173. 
Lobatschewsky,    Nicholaus.      Geometrical    Researches,  on    the 

Theory   of    Parallels ;    tr.    by    G.    B.    Halsted.      Austin, 

Texas,  1892  (date  of  dedication).     104. 
Lucas  Edouard.    Recreations  Mathematiques.   4  vol.    Gauthier- 

Villars,  Paris,  1891-6.    *iy,  *7o,  141,  *  1 7 1 ,  186,  *i97,  ^234. 
Lucas,  Edouard.     Theorie  des  Nombres.     17,  22.   • 
McLellan   and   Dewey.      Psychology    of    Number.      Appleton, 

1895-     154. 
McMurry,  C.  A.     Special  Method  in  Arithmetic.     Macmillan, 

1905.     *207. 
Manning,  H.  P.     Non-Euclidean  Geometry.     Ginn,  1901.     107. 
Margarita  Philosophica.     1503.    67  and  frontispiece. 
Mathematical  Gazette.     London.    41. 
Mathematical  Magazine.     Washington.     *20,  40. 
Messenger  of  Mathematics.     Cambridge,  36,  127. 
Monist    (quarterly).      Open   Court   Publishing   Co.     *I9,    186, 

235- 
Napier,  John.     Rabdologia.     1617.    49,  61,  69,  71. 


BIBLIOGRAPHIC  INDEX.  239 

Newton,  Isaac.     Philosophise  Naturalis  Principia  Mathematica. 

1687.     *I49. 
New  York  Education  (now  American  Education).     *2io,  235. 
Open   Court    (monthly).      Open    Court    Publishing    Co.      ill, 

168,  235. 
Pacioli,  Lucas.    Summa  di  Arithmetica. .  .Venice,  1494.    59,  67. 
Pathway  to  Knowledge.     London,  1596.     *68. 
Philosophical  Transactions,  1743.     119. 

Proceedings  of  the  Central  Association  of  Science  and  Mathe- 
matics Teachers.     206. 
Proceedings  of  the  Royal  Society  of  London,  vol.  21.    ,124. 
Public  School  Journal.     *2o6. 
Rebiere.     Mathematique  et  Mathematiciens.     196. 
Recorde,  Robert.     Grounde  of  Artes.     1540.     68 
Richards,  W.  H.     Military  Topography.     London,   1883.     200. 
Row,  T.   S.     Geometric  Exercises  in   Paper  Folding.     Ed.    1, 
Madras,    1893;    ed.   2     (edited  by   Beman   and   Smith). 
Open  Court  Publishing  Co.,  1901.     144. 
Rupert,  W.  W.    Famous  Geometrical  Theorems  and  Problems, 

with  their  History.     Heath,   1901.     124. 
Schlomilch.     Zeitschrift  fur  Mathematik  und  Physik.     ill. 
School    Science   and   Mathematics    (monthly)    Chicago.     *5o, 

90,  125,  159,  206. 
Schubert,   Hermann.      Mathematical   Essays   and   Recreations, 
tr.  by  T.  J.   McCormack.     Open  Court  Publishing  Co., 
1903  (1899).    95,  124,  154- 
Smith,  D.  E.     Rara  Arithmetica.     Ginn,  1907.     234. 
Smith,   D.   E.     Teaching  of   Elementary   Mathematics.     Mac- 

millan,  1905  (1900).    ^56. 
Smith,  D.   E.   The  Old  and  the  New  Arithmetic.     Reprinted 

from  Text-Book  Bulletin  for  Feb.  1905.     Ginn.     *68. 
Stevin,  Simon.    La  Disme  (part  of  a  larger  work).     1585.    59 
Taylor,  J.  M.     Elements  of  Algebra.     Allyn  (1900).     ^'96. 
Taylor,  J.  M.     Elements  of  the  Differential  and  Integral  Cal- 

#   cuius.    Rev.  ed.    Ginn,  1898.     151,  *2i$. 
Taylor,    J.    M.      Five-place    Logarithmic    and    Trigonometric 

Tables.     Ginn   (1905).     40. 
Teachers'  Note  Book  (an  occasional  publication).     *i89. 
Thorn,   David.     The  Number  and  Names   of  the  Apocalyptic 
Beasts.     1848.     181. 


240    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

Thorndike,  E.  L.     Introduction  to  the  Theory  of  Mental  and 

Social  Measurements.     Science  Press,  New  York,  1904. 

♦156-158. 
Tonstall,  Cuthbert.     Arithmetic.     1522.     67. 
Treviso  Arithmetic.     1478.     59,  67. 
Waring,  Edward.     Meditationes  Algebraical     36. 
Widmann,  John.     Arithmetic.     Leipsic,  1489.     162. 
Willmon,  J.  C.    Secret  of  the  Circle  and  the  Square.    Author's 

edition.    Los  Angeles,  1905.     125. 
Withers,    J.    W.       Euclid's    Parallel    Postulate:     Its    Nature, 

Validity,    and    Place    in    Geometrical    Systems.      Open 

Court  Publishing  Co.,  1905.     *I04,  *io5-io6,  107. 
Young.  J.  W.  A.    Teaching  of  Mathematics  in  the  Elementary 

and  Secondary  School.     Longmans,  1907.     *34,  98,  206, 

235. 


GENERAL  INDEX. 


Note:  ^i  means  page  43  and  the  page  or  pages  immediately  follow- 
ing. i49n  means  note  at  bottom  of  page  149.  References  given  in  the 
Bibliographic  Index  (preceding  pages)  are  not  (except  in  rare  in- 
stances)  repeated  here. 


Abel,  N,  H.,   103. 

Accuracy  of  measures,   43L 

Advice  to  a  building  committee, 
201. 

Agesilaus,  55. 

Ahmes  papyrus,    164,  234. 

Al  Battani,    148. 

Alcuin,   234. 

Algebra,   73-103;  teaching  of,  205f. 

Algebraic  balance,  90,  95;   falla- 
cies, 83. 

Alice  in  the  wonderland  of  mathe- 
matics,  218. 

American  game  of  seven  and  eight, 
197. 

Analytic  geometry,  156-157,  226- 
233- 

Anaxagoras,    122. 

Antiquity,  three  famous  problems 
of,    122. 

Apollo,    122,    128. 

Apparatus  to  illustrate  line  values 
of  trigonometric   functions,    146. 

Apple  women,    194. 

Arabic  camel  puzzle,    193. 

Arabic  notation,  52,  66-68;  word 
for   sine,    148. 

Archimedean  proof,   216. 

Archimedes,    i49n. 

Argand,   J.    R.      37,   94. 

Ariadne,    178. 

Aristotle,    83. 

Arithmetic,  9-72;  in  the  Renais- 
sance, 66;  present  trends  in,  51; 
teaching,    54-58;    2osf. 


Arithmetics    of    the     Renaissance, 

66-68. 
Arrangements  of  the  digits,  21. 
Art  and  mathematics,  213. 
Assyria,    164. 

Astronomers,    44,    165,   216-217. 
Asymptotic  laws,    37. 
Autographs    of    mathematicians, 

168. 
Avicenna,    66. 
Axioms  in  elementary  algebra,  73; 

apply  to  equations?  76. 


Babbage,    72. 

Babylonia,    54,    164. 

Balance,    algebraic,    90,    95. 

Beast,  number  of,    180. 

Beauty   in   mathematics,    208. 

Bee's  cell,    118-1 19. 

Beginnings  of  mathematics  on  the 

Nile,    164. 
Benary,    180. 
Berkeley,   George,    150. 
Bernoulli,    88,    168. 
Berthelot,    166. 
Bibliographic   notes,   234;    index, 

236. 
Billion,   9. 
Binomial    theorem    and    statistics, 

159. 
Bocher,   M.,    10311,   21211. 
Bolingbroke,    Lord,    51. 
Bolyai,    10411. 
Bonola,    Roberto,    10711. 


241 


242    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 


Book-keeper's    clue    to    inverted 

numbers,    25. 
Book-keeping,    first     English    book 

on,   68. 
Boorman,    40. 
Brahe,   Tycho,   214. 
Bridges    and    isles,    170. 
Briggs,    50,    165. 
Buffon,    126. 
Building    Committee,    advice    to, 

201. 


Caesar  Neron,    180. 

Cajori,   Florian,   59,    124. 

Calculation,  mechanical  aids,  69. 

Calculus,    149-153,    206,    213,    2241". 

Calculus   of   probability,    124,    126- 
128,    156. 

Camels,    puzzle   of,    193. 

Cantor,    Moritz,    168. 

Cardan,    66. 

Carroll,  Lewis,  201,  218. 

Carus,    Paul,    173. 

Catch  questions,    196. 

Cavalieri,   149m 

Cayley,    140. 

Centers    of    triangle,    133. 

Chain-letters,    102. 

Checking  solution  of  equation,  81. 

Chinese   criterion    for   prime   num- 
bers, 36. 

Chirography    of    mathematicians, 
168. 

Christians  and  Turks  at  sea,    195. 

Circle-squarer's   paradox,    126. 

Circle-squaring,    122-129. 

Circles  of   triangle,    133. 

Circulating     decimals,     11- 16,     40, 
202. 

Cistern   problem,    212-213. 

Civilization    and   mathematics,  217. 

Clifford,    168. 

Coinage,  decimal,   52. 

Collinearity  of  centers  of  triangle, 
133. 

Colors  in  map  drawing,    140. 

Combinations     and     permutations, 
37,    156. 

Commutative  law,   88,    154. 

Compass,  watch  as,    199. 


Complex  numbers,    75,    92;   branch 
of  graph,    231-232. 

Compound  interest,   47. 

Compte,    167. 

Concrete,     mathematics    teaching, 
205,   217. 

Concrete   necessarily   complex,  214. 

Constants     and     variables     illus- 
trated,   152-153. 

Converse,   fallacy  of,   83  f. 

Counters,    games,    191,    197. 

Crelle,   135. 

Crescents  of  Mohammed,    175-176. 

Cretan    labyrinth,    178. 

Criterion    for   prime   numbers,    36. 

Curiosities,    numerical,    19. 


Daedalus,    178. 

Days- work    problem,    213. 

DeKalb    normal    school,    207. 

Decimal    separatrixes,    49. 

Decimalization    of   arithmetic,    51  f. 

Decimals  as  indexes  of  degree  of 
accuracy,  44. 

Decimals   invented   late,    165. 

Declaration  of  Independence,    175. 

Definition  of  multiplication,  98; 
of  exponents,  101;  of  mathe- 
matics,   212-217. 

Degree  of  accuracy  of  measure- 
ments, 43-44. 

De   la   Loubere,    183. 

Delian   problem,    122L 

De  Morgan,  85,  126-129,  140.  166, 
175,    181,    182. 

Descartes,    37,    94,    166. 

Descriptive  geometry,  206. 

Diagonals  of  a  polygon,  174-175, 
216. 

Digits  in  powers,  20;  in  square 
numbers,  20;  arrangements  of, 
21. 

Dimension,  fourth,  143,  223-224; 
only  one  in  Wall  street,    194. 

Diophantus,  37. 

Direction  determined  by  a  watch, 
199. 

Dirichlet,    37. 

Discriminant,    95. 

D'Israeli,    128. 


GENERAL  INDEX. 


243 


Distribution    curve    for    measures, 

156-159. 
Divisibility,  tests  of,  30. 
Division,   Fourier's  method,  23;  in 

first    printed    arithmetic,    67;    of 

decimals,   63,   65. 
Division     of     plane     into     regular 

polygons,    118. 
Divisor,   greatest,    with   remainder, 

194. 
Do  the  axioms  apply  to  equations? 

76. 
Dodgson,  C.  L.,    168,  201,   218. 
Dominoes,  number  of  ways  of  ar- 
ranging,   38;    in    magic    squares, 

187. 
Donecker,   F.  C,   oon. 
Duplication   of   c,ube,    i22f. 


e,  40,  216. 

Egypt,    54,    164. 

Eleven,     tests    of     divisibility     by, 

3i-33. 

English  numeration,  9;  decimal 
separatrix,   50. 

Equation,  exponential,  102;  in- 
solvability  of  general  higher, 
103. 

Equations,  axioms  apply  to?  76; 
equivalency,  77-79;  checking  so- 
lution of,  81;  solved  in  ancient 
Egypt,    164. 

Equations  of  U.  S.  standards  of 
length   and  mass,    155. 

Eratosthenes,    123,   14911. 

Error,   theory  of,   46. 

Escott,  E.  B.,  7-8,  13,  14,  1911, 
32n,   36,   40,   4m,    in,    116,    187. 

Euclid,  103-108,  118,  123,  130,  166, 
202. 

Euclidean  and  non-Euclidean  ge- 
ometry,   104-108. 

Euclid's    postulate,    103-108. 

Euler,  19,  36,  37,  41,  94,  135, 
165,    168,    171,    177,    178. 

Exact  science,   212. 

Exercise    in    public    speaking    210. 

Exponent,  imaginary,  96. 

Exponential    equation,     102. 

Exponents,    10 1,    165. 


Factors,  more  than  one  set  of 
prime,  37;  two  highest  common, 
89. 

Fallacies,  algebraic,  83;  catch  ques- 
tions,   196. 

Familiar  tricks  based  on  literal 
arithmetic,    27. 

Faraday,    166,   217. 

Fechner,   G.  T.,  215. 

Fermat,    i49n,    186. 

Fermat's  theorem,  36;  last  theo- 
rem,   35;    on   binary   powers,    41. 

Feuerbach's   theorem,    135. 

Figure  tracing,    170. 

Fine   art   and   mathematics,   213. 

Forces,    parallelogram   of,    142. 

Formula,   principle   and   rule,    216. 

Formulas   for   prime   numbers,    36. 

Forty-one,  curious  property  of,   19. 

Four-colors   theorem,    140. 

Fourier,    206. 

Fourier's  method  of  division,  23, 
4m. 

Fourth    dimension,    143,    223-224. 

Fox,   Captain,    127m 

Fractions,    54,   202. 

Franklin,    Benjamin,    186. 

Freeman,    E.   A.,    51. 

French    numeration,    9;    decimal 
separatrix,    50. 

Frierson,   L.    S.,    186. 

Fiitzsche,    180. 

Game-puzzle,    191. 

Games  with   counters,    191,    197. 

Gath  giant,   52. 

Gauss,     34,  37,  94,  95,  103,  166,  203. 

Gellibrand,    165. 

General  form  of  law  of  signs,  99. 

General    test   of    divisibility,    30. 

Geometric    illustration    of    complex 

numbers,    92;    of    law    of    signs 

in    multiplication,    97. 
Geometric     magic     squares,      186; 

multiplication,    88,    154;    puzzles, 

109. 
Geometry,    103-145;  teaching,  205f ; 

descriptive,    206. 
German     numeration,     9;     decimal 

separatrix,    50. 
Giant  with  twelve  fingers,   52. 


244    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 


Girard,    Albert,    37,    92. 

Glaisher,    39,    7111,    167. 

Golden    age    of    mathematics,    203. 

Gotham,  square  of,  189. 

Grading  of  students,    159. 

Graph    of    equation,    156-157,    227- 

233- 

Gravitation,  215. 

Greatest  divisor  with  remainder, 
194. 

Greeks,  37,  54,  56,  66,  72,  123, 
148,    i49n,    167,   186,  216,  217. 

Growth  of  concept  of  number,  37. 

Growth  of  philosophy  of  the  cal- 
culus,   149. 

Gunter,    165. 

Hall,   W.   S.,    158. 

Halsted,   G.   B.,    104. 

Hamilton,    W.    R.,    94,    168. 

Hampton    Court    labyrinth,    178. 

Handwriting  of  mathematicians, 
168. 

Heron  of  Alexandria,   212. 

Hexagons,  division  of  plane  into, 
118;   magic,    172-173,    187-188. 

Hiberg,    149m 

Higher    equations,    103. 

Highest  common   factors,   two,   89. 

Hindu  check  on  division  and  mul- 
tiplication, 25;  illustration  of 
real  numbers,  91,  92;  numerals 
(Arabic),  52,  66-68;  word  for 
sine,    148. 

Hippias  of  Elis,    123. 

History  of  mathematics,  167;  sur- 
prising facts,    165. 

Hitzig,    180. 

Home-made   leveling    device,    120. 

Ideal,  mathematics  science  of,  214, 

215. 

If  the  Indians  hadn't  spent  the 
the  $24,   47. 

Illustrations  of  the  law  of  signs, 
97;  of  symmetry,  144;  of  trigo- 
nometric functions,  146;  of  lim- 
its,   152. 

Imaginary,    94;      exponent,    96; 
branch   of   graph,    230-232. 

Indians    spent    the   $24,    47. 


Infinite,     87,     224f;     symbols     for, 

151- 
Inhabited  planets,   216. 
Inheritance,    Roman   problem,    193. 
Instruments    that    are    postulated, 

130. 
Interest,  compound  and  simple,  47. 
Involution    not    commutative,    154. 
Irenaeus,    1 80-181. 
Isles  and  bridges,   170. 
Italian     numeration,     9;       decimal 

separatrix,    50. 

Jefferson,   Thomas,    175. 

Kant,    167. 

Kegs-of-wine    puzzle,     194. 
Kempe,   A.   B.,    132,    136,    139. 
Kepler,    50,    107,    i49n,    167,    203, 

214,  217. 
Kilogram,   155. 
Knilling,    57. 
Knowlton,   197. 
Konigsberg,    170- 171,    174. 
Kiihn,    H.,    93,    94. 
Kulik,  40. 

Labyrinths,    170,    176-179. 

Lagrange,    36,    168. 

Laisant,    166. 

Laplace,    126,    168. 

Lathrop,    H.   J.,    145m 

Law  of  signs,  97;   of  commuation, 
154- 

Legendre,    36,    37,    168. 

Lehmer,    D.   N.,   40. 

Leibnitz,     149-150,     166. 

Length,   standard  of,    155. 

Lennes,   N.   J.,   90m 

Leonardo   of   Pisa,    66. 

Leveling   device,    120. 

Limits    illustrated,    152. 

Lindemann,  '  123,    124. 

Line  values  of  trigonometric  func- 
tions,   146. 

Linkages  and  straight-line  motion, 
136. 

Literature   of   mathematics,    203, 
208-209. 

Lobachevsky,     104-108. 


GENERAL  INDEX. 


245 


Logarithms,     45,    47,    52,    69,    87, 

102,    165.      See  also  e. 
London   and  Wise,    176. 
Loubere,   de  la,    183. 
Lowest    common    multiples,    two, 

89. 
Loyd,    S.,    116,    18711. 
Lunn,   J.   R.,   40. 
Luther,    181. 

Maclaurin,    119. 

Magic  number,  25;  pentagon,  172- 
173;  hexagors,  172-173,  187- 
188;    squares,     183. 

Manhattan,  value  of  reality  in 
1626   and   now,    47-48. 

Map  makers'  proposition,    140. 

Marking    students,    159. 

Mars,    signaling,    216-217. 

Mass,   standard  of,    155. 

Mathematical  advice  to  a  building 
committee,  201. 

Mathematical    game-puzzle,    191. 

Mathematical  reasoning,  nature  of, 
212. 

Mathematical  recitation  as  an  ex- 
ercise in  public  speaking,  210. 

Mathematical    recreations,    234. 

Mathematical    symbols,    162,    165. 

Mathematical  treatment  of  statis- 
tics,   156. 

Mathematics,  definitions,  2i2n; 
nature  of,  212-217;  teaching 
more  concrete,  205,  217;  Alice 
in  the  wonderland  of,   218. 

Mazes,    176-179. 

Measurement,  numbers  arising 
from,   43 ;    degree  of   accuracy 

of,    43-44- 

Measurements  treated  statistic- 
ally,   1 56- 161,    207-208. 

Measures,    standard,    155. 

Mellis,  John,  68. 

Methods  in   arithmetic,    54-58. 

Metric   system,    43,    53,    155. 

Miller,    G.   A.,    50. 

Million,  first  use  of  term  in  print, 
67. 

Minotaur,   178. 

Miscellaneous  notes  on  number, 
34- 


Mobius,  A.  F.,   140. 

Mohammed,    175-176. 

Morehead,  J.    C.,   41. 

Moscopulus,    186. 

Movement  to  make  teaching  more 
concrete,   205. 

Multiplication  at  sight,  15;  ap- 
proximate, 45,  62,  64;  of  deci- 
mals, 59;  in  first  printed  arith- 
metic, 67;  law  of  signs  illus- 
trated, 97;  definition,  98;  as  a 
proportion,  100;  gradual  general- 
ization   of,    100;    geometric,    88, 

154- 
Myers,  G.  W.,  9on. 

11  dimensions,  104.  See  also  Fourth 
dimension. 

Napier,  John.      See  Logariths. 

Napier,    Mark,    165. 

Napier's  rods,   69. 

Napoleon,    167. 

Nature  of  mathematical  reason- 
ing,  212. 

Negative  and  positive  numbers,  90. 

Negative  conclusions  in  19th  cen- 
tury,   103. 

Neptune,    distance   from   sun,   44. 

Nero,    180. 

New   trick   with    an   old   principle, 

15- 

New  York,  value  of  realty  in  1626 
and  now,   47-48. 

Newton,   49,    149-150,   215. 

Nicomedes,    123. 

Nile,  beginnings  of  mathematics 
on,   164. 

Nine,   curious  properties  of,    25. 

Nine-point    circle,    134-135. 

Nineteenth  century,  negative  con- 
clusions reached,    103. 

Non-Euclidean    geometry,    104-108. 

Normal    probability    integral,     157. 

Number,  miscellaneous  notes  on, 
34-42;  growth  of  concept  of,  37; 
How  may  a  particular  number 
arise?  41;  of  the  beast,   180. 

Numbers  arising  from  measure- 
ment, 43;  differing  from  their 
log.  only  in  position  of  decimal 
point,    19;   theory  of,   34L 


246    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 


Numeration,    two    systems,    9. 
Numerical  curiosity,   19. 

Old-timers,    194. 

Oratory,  mathematical  recitation 
as  exercise,    210. 

Orbits   of    planets,    214. 

Oresme,    10 1. 

Orthotomic,  94.  See  also  Imag- 
inary. 

Oughtred,   49. 

7T,  -40,    123-129,    216;    expressed 
with  the  ten  digits,   23. 

Pacioli,    59,    100. 

Paper  folding,    144. 

Paradox,  circle-squarer's,   126. 

Paradromic    rings,    117. 

Parallel    postulates,    103-108. 

Parallelogram  of  forces,    142. 

Parallels  meet  at  infinity,    107. 

Peaucellier,    136-139. 

Pentagon,    magic,    172-173. 

Periodicals,  publication  of  fore- 
going sections  in,   235. 

Permutations,    37,    156. 

Petzval,   40. 

Philoponus,   122. 

Philosophy  of  the  calculus,    149. 

Pierpont,   James,   203. 

Pitiscus,    49,    50. 

Plane,  division  into  regular  poly- 
gons,   118. 

Planetary    orbits,    214. 

Planets,    inhabited,    216. 

Planting  in  hexagonal  forms,    119. 

Plato,  122,   123,   130,   166,  211,  214. 

Plato  Tiburtinus,    148. 

Positive  and  negative  numbers,  90. 

Powers   having   same   digits,    20. 

Present  trends  in  arithmetic,  51. 

Prime  factors  of  a  number,  more 
than  one  set,  37. 

Primes,  formulas  for,  36;  Chinese 
criterion   for,   36;    tables  of,    40. 

Principle,    rule    and   formula,    216. 

Probability,   124,   126-128,   156. 

Problems  of  antiquity,  122;  for 
quickening  the  mind,   234. 

Products,   repeating,    11- 16. 

Proportion,   multiplication   as,    100. 


Psychology,  54,  57,  215. 
Ptolemy,  103,  167,  217. 
Publication    of    foregoing    sections 

in  periodicals,  235. 
Puzzle,  game,    191;  of  the  camels, 

193- 
Puzzles,    geometric,    109. 
Pythagorean  proposition,    121,   164. 

Quadratrix,    123. 

Quadrature  of  the  circle,    122-129. 
Quaternions,  88,  94,    154. 
Question    of    fourth    dimension   by 

analogy,    143- 
Questions,   catch,    196. 
Quotations   on   mathematics,    166. 

Real   numbers,    90,    232. 

Reasoning,   mathematical,   212. 

Recitation   as  an   exercise   in   pub- 
lic speaking,   210. 

Recreations,    mathematical,    234. 

Rectilinear    motion,    136-139- 

Recurring  decimals,    11-16,   40. 

Regular  polygons,  division  of  plane 
into,    118. 

Reiss,  38. 

Renaissance,    arithmetic    in,    66. 

Renaissance    of    mathematics,    203. 

Repeating    decimals,     11-16,    40; 
products,    11-16;   table,    17. 

Representation    of    complex    num- 
bers,   92,   231-232. 

Reuss,   180. 

Riemann's   postulate,    105-108. 

Rings,    paradromic,    117. 

Rods,    Napier's,    69. 

Rornain,    Adrian,    6of. 

Roman    inheritance    problem,     193- 

Roots   of   equal    numbers,    73,    75; 
of   higher   equations,    103. 

Rope   stretchers,    121. 

Royal    Society's   catalog,    203. 

Rudolff,    162. 

Rule,    principle   and   formula,    216. 

Ruler  unlimited  and  ungraduatcd, 
130-132. 

Scalar,    94.      See    also    Real    num- 
bers. 
Scheutz,    72. 


GENERAL  INDEX. 


247 


Separatrixes,  decimal,  49. 
Seven-counters  game,  197. 
Seven,  tests  of  divisibility  by,   31- 

33- 
Shanks,    William,    40,    124. 
Signatures  of  mathematicians,  168- 

169;   unicursal,    170,    175-176. 
Signs,   illustrations  of  law   of,   97. 
Sine,  history  of  the  word,  148. 
Smith,   Ambrose,    127. 
Smith,    D.    E.,    56,    59,    168,    234n. 
Smith,   M.   K.,    159. 
Social    sciences   treated    mathemat- 
ically,   156,   207-208. 
Societies'   initials,   38. 
Sparta,    55. 

Speaking,    recitation    as    an    exer- 
cise in,   210. 
Speidell,    165. 
Square   numbers   containing   the 

digits  not  repeated,   20. 
Square  of  Gotham,    189. 
Squares,    magic,     183;    geometrical 
magic,    186;    coin,    187;    domino, 
187. 
Squaring  the  circle,    122-129. 
Standards  of  length  and  mass,  155. 
Statistics,    mathematical    treatment 

of,    156,    207-208. 
Stevin,   Simon,   59f,    10 1. 
Stifel,   91. 

Straight-edge,    130-132,    136. 
Straight-line   motion,    136. 
Student   records,    159. 
Shuffield,   G.,  40,   41. 
Surface  of  frequencey,    156-159. 
Surface  with  one  face,    117. 
Surprising   facts  in   the   history  of 

mathematics,    165. 
Swan  pan,   72. 
Sylvester,    J.    J.,    139,    168. 
Symbols,    mathematical,    162,    165; 

for   infinite,    151. 
Symmetry    illustrated    by    paper 
folding,    144. 


Tables,   39;   repeating,    17, 
Tait,    19. 
Tanck,    57. 
Tax  rate,  46. 


Taylor,   J.    M.,    7. 

Teaching  made  concrete,  205f,  217. 

Terquem,    135. 

Tests  of  divisibility,   30. 

Theory  of  error,   46;   of  numbers, 

34*. 
Theseus,    178. 
Thirteen,    test    of    divisibility    by, 

32. 
Thirtie   daies   hath   September,   68. 
Thirty-seven,    curious   property    of 

19. 
Three    famous    problems    of    anti- 
.    quity,   122. 

Three    parallel    postulates    illus- 
trated,   105. 
Time-pieces,    accuracy   of,    43. 
Trapp,    57. 

Trends  in  arithmetic,    51. 
Triangle   and   its   circles,    133. 
Trick,   new  with   an  old  principle, 

15- 
Tricks  based  on  literal  arithmetic, 

27. 
Trigonometry,     96,     107,     146-148, 

165. 
Trisection  of  angle,    i22f,   130-132. 
Turks  and  Christians  at  sea,    195. 
Two  H.  C.  F.,  89. 
Two   negative   conclusions   reached 

in  the   19th  century,    103. 
Two  systems  of  numeration,   9. 
Tycho  Brahe,   214. 

Undistributed   middle,    83f. 
Unicursal    signatures    and    figures, 

170. 
United  States  standards  of  length 

and  mass,    155. 

Variables  illustrated,    152-153. 

Vectors,   88,   94,    154. 

Vienna  academy,   40. 

Visual    representation    of    complex 

numbers,    92. 
Vlacq,    165. 
Von   Busse,    57. 

Wall   street,    194. 
Wallis,   93,    101,    151. 
Watch   as  compass,    199. 


248    A  SCRAP-BOOK  OF  ELEMENTARY  MATHEMATICS. 

Wonderland   of   mathematics,    218. 


Weights  and  measures,  43,  53,  155. 

Wessel,    37,   94. 

Whewell,    167,   217. 

Wilson,  John,  biographic  note,  35m 

Wilson's  theorem,    35. 

Withers,  J.   W.,    107. 

Witt,    Richard,   49. 


Young,    J    .W.    A.,    205. 

Zero  in  fallacies,  87,  meaning  of 
symbol,  1 50-1 51;  first  use  of 
word  in  print,  67. 


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